You definitely should buy your own if you go climbing a lot (at the very least, if you assume a $90 pair and $3 rentals, even going once a week only takes a few months before it makes economic sense).
That said, they aren't exactly comfortable. They shouldn't be painful
to use, but they should be quite snug and probably a bit uncomfortable -- at least to the extent that you wouldn't want to wear them casually. This does a couple things. First, it makes sure your feet don't slide around inside the shoe. Second, it will allow more of the feeling of the rock to be transferred to your feet. Third, it will allow your toes to reinforce each others' strength, so you're not, e.g., standing on essentially just
your big toe.
That said, if you're a beginner (and considering you don't have your own shoes...) you should probably err on the side of comfort. If your shoes make you want to not climb, that's no good at all. You also shouldn't go out and spend a fortune on shoes unless you've got money burning a hole in your pocket*. Three reasons: 1) Beginner (read: bad) footwork will wear them faster, 2) you won't have as good of an idea of the style you want to climb (and thus what style shoe is best), and 3) you'll be doing easier routes where it doesn't make as much of a difference. As you improve you'll go "I should get a new pair of shoes" and then you'll probably get a much tighter pair. (I've heard someone say that your second pair of shoes is probably as good of a benchmark as any of an intermediate climber.
* On second thought, if you have money burning a hole in your pocket you should still get cheap shoes and send me the difference.
Also: non-synthetic shoes (most shoes, other than those by the brand Evolve) will stretch a bit, maybe a half size. My first pair are absolutely enormous now. (Also, don't overcompensate on the second pair: I did, and I stopped using them and got a 3rd pair because they were too small.)