As a man, I still consider it derogatory and would not use it in any normal conversation. I will agree that it's quite mild as a term, not a major insult, but it's still derogatory.
It's originally a slang term, derived (if I'm not mistaken) from the older British slang of "bird" for a woman. (Thus, young woman = "chick".) It was almost invariably used between men, with the "bird" being a form of quarry to be hunted, and the term still has that connotation. It is definitely dismissive; the person referred to that way is not on an even status with the speaker.
As far as groups that refer to themselves with it, such as the cited "Dixie Chicks" and "Chicks with Picks", I consider them to have understood the connotations and intentionally used it ironically. I'll refer to the organization as they prefer, but personally I still wouldn't walk up to a random member of "Chicks with Picks" and say, "Hey chick, come over here." At least, if I did I wouldn't expect a positive reaction. :)
I think it's probably just because in British slang pull has that meaning independently of bird. From OED...
pull: trans. 12a: Brit. slang. To pick up (a partner), esp. for sexual intercourse; to seduce. Also intr.
It also occurs as a noun in the expression on the pull, and there's no reason why a couple of young British men shouldn't hope to pull some girls on a night out. Come to that, the girls they end up with may have gone out hoping to pull some blokes.
As to why young British women are called birds, OED says it derives in part from a now-obsolete
burd: a poetic word for ‘woman, lady’; the female counterpart of berne n.;
in later use chiefly = ‘young lady, maiden’.
berne: a warrior, a hero, a man of valour;
in later use, simply one of the many poetic words for ‘man’.
Of chick, OED says applied to human offspring; = chicken n.; esp. in alliteration with child. Sometimes as a term of endearment, with citations starting from 1320. But their earliest citation for the current (well, hopelessly "dated", imho) sense girl; young woman. slang (orig. US) is 1927.
Best Answer
Well only a woman can be a bird. Yes it's similar to 'chick' in the US. It's not a particularly offensive word but you wouldn't use it in any sort of polite company or probably when women were present. In fact you probably wouldn't use it at all as it sounds somewhat dated to the 70s and 80s.