Bestämma sig is a reflexive verb that requires a pronoun afterwards to further show who the subject is. Examples: Jag bestämmer mig (I decide), Du bestämmer dig (You decide), Han bestämmer sig (He decides), and Vi bestämmer oss (We decide). Some other reflexive verbs are känna sig (to feel) and förbereda sig (to prepare).
In Vi försöker bestämma oss, I think it means more specifically "to make up our minds." In Hon vill alltid bestämma, I think it means that she wants to decide as opposed to making up her mind. To make that a bit more clear, I think that bestämma is reflexive when you are deciding something for yourself that will affect only your actions. I'm not 100% sure, though!