Can medications for depression affect who you are as a person?

They Shouldn't. The nature of depression medications is to alter the amounts or effectiveness of neurotransmitters in your brain. If you are taking an anti-depressant and you feel "off" or not like yourself, talk with your doctor. Consider trying a different medication. There are enough options that you should be able to find treatment that helps your depression without making you feel like a different person.
Perhaps. Some who have been very depressed can feel liberated when an antidepressant begins to work. Peter kramer's "listening to prozac" patients felt "better than well, " even. On the other hand, i've known people who tell me, "i'm no longer suicidal, but i don't feel like myself either." sometimes this may be the unfamiliarity of no longer being depressed, but in some cases it can be a "dulling" effect.