There are many ways I could start writing about Rose's personality and the way she interacts with fellow members of the group, but let's do it with this since it's been a long time in the making:
That's Rose during the first two chapters, as in the first half of the story before the quest becomes more urgent and the mysteries deepen; the end of disc 2 is a turning point – in many ways, but especially as far as story exposition and characterization is concerned. As you see, Rose likes to keep her distance: Pretty much every time the party stops to take a rest somewhere during the first half of the game, she chooses a nice corner and a wall to casually lean against while observing the happenings from the back, occasionally contributing information or making cynical to ominous remarks, the weight of which doesn't hit you until much later. Even outside of towns and cities, she's always standing off to the side or at the far back, bringing up the rear. One detail I adore about The Legend of Dragoon is how this behaviour is captured in the official art (the promotional images) of the game, where Rose continues to skulk in the background of group shots – always present, but not necessarily fully visible. Rose doesn't take kindly to being spoken to in an overly familiar way, or to being touched: To Haschel, she firmly says "Don't 'tootsie' me.", and she instantly knocks the kind Lavitz over when he tries to drag her away at one point to give Dart and Shana some alone time – "(Why do I have to go with you?)".
More subtle is the fact that along with physical distance, Rose also remains emotionally aloof. On the one hand, Rose doesn't get as involved with the group on an emotional level – yes, she speaks with Dart, yes, she explains things to them, but she's not close to anyone. This is a stark contrast to the way Dart draws people to him, the way everyone is worried about Shana, how Lavitz' kindness and passion affect those around him, or – perhaps the most obvious contrast – how easily Meru befriends others with her sociable nature, physically and emotionally. The several times that the physically frail Shana holds up the party (twisted ankle, susceptibility to poison, actually getting poisoned, collapsing, etc.), party members present rush to her side or watch over her, but Rose always stays back and makes no effort to help or to show her concern in some way. When Lavitz receives the news that his king has been abducted and loses control, Dart and Haschel have to restrain him; Rose, however, looks on impassively. At one point, everyone in the group tries to keep themselves occupied so as not to be overcome by emotion while staying close to one another, but again Rose alone watches from the sidelines. Is Rose just plain cold and uncaring, does she not know how to express that she cares or is she actively trying to stay uninvolved, afraid to get attached? That is a question to revisit at a later point.
On the other hand, this kind of emotional distance and indifference is reflected in her composure. Rose is rarely surprised: not when Shana accuses her of causing Dart's loss of consciousness, not when they meet multiple creatures that shouldn't exist in this age anymore, not when they fall through a trapdoor, when they are ambushed by bandits or arrows are shot at them unexpectedly – unlike Dart and Lavitz, for example, the latter being an experienced knight. And when she is surprised, as is the case when more Dragoon Spirits acknowledge members of the group, she keeps it to herself. It is perhaps this composure that makes others within the group look to her for guidance, for even though they don't know much about their companion, they can tell that she's a veteran. Rose's confidence in her skills shows in and outside of battles without her being arrogant. Party members look up to her in different ways: Shana, for example, tries so very hard to contribute to the group (perhaps too hard), especially after meeting and comparing herself to Rose, while Lavitz trusts her very quickly and acknowledges her skill.
Shana: I felt a little bit dizzy, that's all.
Lavitz: I guess it is hard for a woman.
Rose: Oh, I am a woman too.
Lavitz: You are special.
Rose: "Special", huh? Fine. I don't capitalize on a woman's frailty anyway.
Shana: I don't mean to do... oh no. I'm okay. See? Rose is fine too. But, I... I'm sorry. I need to go rest a little. Disc 1: Nest of Dragon
As an experienced warrior, Rose always remains focused on the task at hand; the other side of this drive is her no-nonsense attitude and her impatience. I didn't remember that last aspect from my first playthrough, but it was very noticeable this time around: Rose may be pragmatic and goal-oriented, but she's also the sole member in the group who, in a way, keeps rushing the rest by making her disapproval or indifference known, vocally or physically. When Shana twists her ankle and the party takes a break, Rose sighs and makes an impatient gesture behind their backs; the two times that they are ambushed by bandits, once due to a misunderstanding, once with clearly ill intentions, Rose plainly states it's easier to resolve the matter by beating them into submission; furthermore, there are multiple times she explicitly tells the party to stop wasting time talking and to make up their mind. This is not to say that this is necessarily received negatively by the group, as they certainly can use someone who keeps them on the right track.
Rose also has very little patience for people outside of the party. She doesn't appreciate time being spent on small talk with villagers when something needs to be done, and does not care much about the feelings of those who happen to cross the group's path. A memorable example of this is at the end of disc 1, where they meet a child belonging to a group that helps people get through hard times during war. When the little boy sees the king, he expresses his regained hope that the war will soon find an end so that he can meet his parents again. And how does Rose react? "Don't waste time babysitting like that. Why don't you think about how to get into the castle?", followed by a decisive "I'm going ahead." when the boy conveys just how much he is moved to find out the king is still alive and all party members vow to put an end to the war. Evidently, Rose neither has patience for pretty words nor any sympathy for those around her.
This impatience strongly plays into her bluntness, as she doesn't indulge others and tends not to speak much either. Rose has that remarkable ability to often state what people least want to hear in a dire situation instead of offering comfort ("It's our own fault for being trapped.", "What is a king to do if he doesn't have a castle to go back to and a large force to protect him?", etc.); Haschel in particular calls her out on this more than once:
Shana: It all started with the Dragoon Spirit.
Dart: We have to get it back quick or it'll be too risky.
Rose: You shouldn't have let it be taken to start with, then.
Haschel: You are harsh as always. There are some other ways to say that. Why don't you show a little smile at least?
Rose: I don't remember... how to smile. Good night. Disc 2: Twin Castle at Fletz
Unsurprisingly, she brushes it off just as she brushes him off when he brings up her impatience in a personal conversation later on (which deserves a separate page on the shrine): "It's none of your business."
Lastly, Rose can be a scarier foe than anyone in the party as she's swift, deadly and merciless; just look at her threaten various enemies over the course of the game (that translation work!):
"This is your last chance. Get out of our way or you'll need a real doctor."
"Screams won't be heard down there... Why don't we settle this here?"
"I'm sorry but you don't have time to be in love anymore. Because you will die here."
This takes on a darker shade when you recognize the black and white thinking she shows in certain situations, especially in matters of life and death. Here are two separate encounters with Kongol, a Gigantos (the species that sets itself apart with its enormous size and brute strength) who fights for the enemy:
Kongol: ARRRGGHHH! Gigantos, with strongest armor... lost. Me, brother of hero... has pride. Cannot live in disgrace... Kill me.
Rose: As you wish.
Dart: Stop it! The game is already over!
Rose: "Game"? Don't be silly, this is a matter of life or death.
Shana: It's over. He cannot fight.
Rose: Hope your sweetness won't kill you. Disc 1: Black Castle at Kazas
Kongol: Emperor Doel say he make world where every species equal. World needs strong leader. Whether you can be leader, Kongol will watch to the end. Gigantos' sadness should not be repeated...
Rose: There is no such world where "equal" exists. What we have is killing and being killed. That's it. Disc 2: Home of Giganto
As a closing remark, it's worth noting that pretty much everyone in the party generally maintains a respectful distance from Rose (the way she prefers it to be); it wouldn't be too far off to say that at least some of them are slightly intimidated by her cold exterior as well. Nevertheless, everyone trusts and appreciates Rose as a companion on a lengthy journey:
Dart: (Don't play a prank on her. She doesn't understand jokes.)
Meru: (I won't! Rose has a mysterious personality. She is always cold but when it comes to a critical point, she leads us in a variety of ways.)
Dart: (We are alive now because of Rose.)
Meru: (I agree.) Disc 2: The Queen Fury
All in all, when Meru mutters "[Rose is dark... Something is totally wrong with Rose...]" to herself, one can only agree (and marvel at her accurate assessment).