I think the argument given by 5xum is the basic reason we trust most proofs. However, I know of several scenarios where circular proofs could possibly get into the literature.
One scenario is a mathematician writing a number of papers published in parallel. If the papers cite each other, there could be a circularity. A prominent mathematician (a Fields medalist) once told me of a case of an apparently circular argument in a series of papers by a famous mathematician. He believed it was actually a deliberate deception. Since I haven't personally verified this, I'm not going to name names. While this is possible, it is a rather rare circumstance.
Another scenario is theorem A is published. Later theorems B, C, ... are proven based on theorem A. Later, other proofs of theorem A are given possibly based on some of the theorems B, C, ... Such later proofs are circular and invalid, but at least the original proof is valid, so we may have invalid proofs but at least the theorems are all correct. Except ... what if the original proof of theorem A was erroneous and no one discovered the flaw? Or, even if the flaw in the original proof was discovered, but the later proofs are still accepted as valid. It may be very hard to sort out what is valid and invalid if there's a lot of intervening work.
Similar things can happen when textbooks are written that cite each other. I know of cases of erroneous statements given as "theorems" in well regarded and frequently cited texts. Often, such "theorems" are left as exercises or are claimed to be easy to prove.
So, ..., I generally trust the mathematical literature, but it's quite possible there are circular proofs out there. I also think there are many undiscovered errors in published proofs. Anyone who's done any computer programming knows that bugs are unavoidable and sometimes maddeningly difficult to find, even in seemingly simple and clear code. I have no doubt the same is true of math, except we don't have to pass our proofs through compilers and run them.
In the long run, I think the answer will be that some day we will run our proofs through compilers, in the form of automated proof verifiers. Gradually, we'll codify all of mathematics into machine verifiable systems that won't allow circular proofs.