The closing is done by whoever you specify in the contract, but if you use the HUD closing agent then HUD will pay their closing fee and quite frankly if you don't use their closing agent they may whine about how you should have used their agent (I'd say like 90% or more HUD closings use the HUD closing agent). Though HUD will close with an agent you choose if you make them. On HUD's behalf, they seem to want you to use their agent as their agents work closely with them and are pro's at closing these types of deals, use an agent who hasn't done these and they may mess stuff up - so it really is also in your interest to use their closing agent.
At the closing the buyers (you guys) will need to be there and your agent will probably want to be there more or less to explain stuff and "hold your hands" through the process (though they don't need to be). That's it, nobody from HUD will be there, and though the bank could choose to have someone there they almost certainly won't bother, though if you use a mortgage broker they might come, more or less as a ploy to say "see what a great mortgage broker I am" - they don't need to be there either.
You typically close on your own with the HUD designated closing agent (or whoever you choose to close it if you did designate someone in the contract) and then HUD signs all the closing documents later.
I've closed lots of these and never really had a problem after the fact (like the closing agent forgot something and HUD wouldn't close the sale) but, as mentioned, since HUD won't be there thats another good reason to use their closing agent (HUD closings are a bit different but if you use their closing agent they have probably closed literally 1000's of these so they know the process inside and out).