You aren't feeling down from the prozac, you would need to be on them for about 3 weeks to notice any mood improvment anyway. You shouldn't take anyone else's meds. SSRI's are a class of meds that need to be taken over a period of time and often titrated for any results, if they are for you anyway, not all meds work for everyone, you need a pdoc to be the best judge of what you should do.
It isn't un natural to feel affected by someone close to you when they aren't well or depressed, but if this low feeling is interferring with your daily personal schedule, functioning, or you just do not feel like yourself, it may be a good idea to take it up with a mental health care professional, maybe even start with a counselor.
Don't let this go too much longer, there is help out there, life is too short to feel crappy. . . enjoy it.
I wish you luck with this, and I feel confident you'll get along fine, just do not "share" meds with others, see a professional if you are to be put on meds.
Please take care,
DE

In giving advice seek to help, not please your friend
SOLON