You can omit the topic (あなた/ANATA) of the sentence when it is understood. If it is just you and a Japanese person, you can just ask her/him なんねんうまれですか (NAN NEN UMARE DESUKA/What year were you born?).
If there are more people around you, it is better not to omit the topic of the sentence. You can either use あなた (ANATA/you) or their own names as in the example Q&A. Actually it sounds more friendly if you use their names.
I've had people tell me the same. One time I was attempting a conversation and we both used あなた and the other Japanese guy who was there says "Why are you guys talking like that? You sound like husband and wife!"
If I have to I still use あなた、but I try to avoid it. If I know the person's name, I stick with that さん。Also, when talking to female friends who are younger than me and I know very well, I use きみ often for "you". It seems to go over well.
Thanks for the asnwers To follow on from this I presume that you would use せいねんがっぴ は いつ でした か to ask when someone was born ie the date. (expecting to be told the date, and year)? And in this case would it be right to use でした ?
limited usage It isn't very common to ask someone "what is your date of birth?". It is more normal to say たんじょうびは いつですか。 And just one more issue you wouldn't use でした either since it isn't a past tense thing. Sure their date of birth was in the past but it is a current fact so you would use です in that sentence, if you were to use that sentence, although I don't recommend it.
I guess having being part way through my partner being treated in hospital I am used to people asking when his date of birth is. Quite a few companies use it for a security check too. Is this not the case elsewhere?