Assuming you don't have a trade-in to worry about (where they could try to make up the difference), you still probably need to see the final bottom-line price before you know whether it's as good as it seems.
I assume they'll charge you a processing fee and maybe an advertizing fee. Around here that adds another few hundred $.
Those prices do seem pretty low. The cheapest 2005 Camry LE I've seen in browsing online at carmax.com is $18k - $700 rebate). But for some reason, the price on Corollas didn't really change much in the 1990s (my wife's 1993 and her parents' 1999 cost about the same - ~ $12k), so maybe you can get one for $13k.
How does the salesman seem otherwise? If s/he seems slimy it's unlikely that s/he'll be doing you a favor. If s/he seems great, s/he may still be sharpening a shiv to stick in your back...
It might be worth it to get several quotes via Edmunds.com or a similar site. How's the dealer reputation?
Some people do get good deals in their car shopping. Maybe he likes you or is terrified of you. ;-)
Luck!
Cheers,
Scott.