As long as your not doing anything illegal it really shouldn't matter if you give them your ID or not.
of course, but becuase of the stigma around bitcoins I would prefer to avoid unwanted attention from the "authority." I guess that would be my concern.
Ok, then don't use a bank transfer. Localbitcoins is a great way to get bitcoins, you will be paying a higher premium though, so that's the other side to consider. You could also try buying giftcards (such as WalMart) and selling them on the forums, although it could be tough with a newbie account as trust is a pretty big factor.
EDIT: BTW, the only time the authorities would get your information is if there was a need for it. Just because the bank takes your ID, doesn't mean they send it to any government agency. They simply do it incase the government comes asking (which in my opinion, if your just transferring money to an exchange or something, there is a very very almost zero small chance it would be an issue.)