This is just my opinion...
Ok giving a girl your facebook/myspace or any social networking site....
Pro's -
Can chat and get to know girls better
Can check out her pics, see it was just a one off that she looked good
Con's -
The minute she is on you FB acount you have taken away that "Mystery" air that you worked hard to build up when u first met her. She can see you at your best and worst..... this is a massive down side
so I know I have nowhere near covered all of the angles on this because to be honest, i dont ever give out my fb to girls im gaming (unless ofcourse they wer already friends)... I just see that the control that you lose by giving girls you fb details is too much to give away.
When all they have is a phone number, you have all the control, how long you leave it to get in touch, what they find out about you, etc.... im currently gaming a girl, she said
Her: whats your second name?
Me: you just want my surname so that you can stalk me on facebook.
Her: Yeah maybe

Me: I thought you were cool? Don't you ever take a risk? haha
Her: Hmm we'll see
Final thought:
If a girl has seen everything on your facebook, all your friends comments, all your pics, all your hobbies, everything..... what else is there for her to find out about you? There is nothing she can find out about you in person that she cant just with a click of her mouse..... surely we should value ourselves more than that? If a girl wants to find out about me... shes gonna have to come out for a drink and ask me questions, thats part of the excitement of a date.