Same solicitor, since it'll be a single transaction. I very much doubt, in fact, whether you'd be able to get separate solicitors to act for you.
It's advisable, as well as agreeing division of costs, etc., to have some sort of agreement in place on what happens if you break up after a relatively short time - does one of you have the right to buy out the other (if so, which), do you automatically sell, that kind of thing. Naturally you never expect to be in the position - otherwise you wouldn't be making this big step! - but it's better to agree something amicably rather than have to deal with it in an emotionally traumatic situation later.
Best of luck!