Do both sites rank similarly for the same keywords? Is one stronger than the other. Are both sites unranked? Does example2.com have backlinks that point to internal pages? Answers to those questions would help determine what would be your best course of action.
If example2.com has decent back links and pagerank, why not keep the two separate and utilize example2.com for a little link juice to example1.com and for reputation management purposes?
If you were to keep them separate, the goal, to me, would be to keep the two domains completely untied to each other except for a well placed link or two. You can revise contact info, company names, and forms on example2.com if you wanted to but leave the existing page names as they are so backlinks retain continuity and maximum effectiveness.
Private registration could be a good idea since Google will be able to determine that you are the new owner of example2.com and that you are also the owner of example1.com.
If you were to do that, you would keep example2.com on it's original IP address at it's original host. If not, then move it to an IP address at a third host. If not that, then move it to a non-shared IP address at your own host.
That would be one way of handling it. Of course 301ing everything over to example1.com would be the easy way of handling it.