I have an FSA not an HSA (sorry about that - realized you were talking about the other type of account - we don't have that one available), and yes, the company that manages the account takes care of the paperwork. We never have any money left in the account to roll over because we have huge medical bills due to my husband's medical condition (I would love to have that problem). I don't claim additional medical expenses on my tax return (wouldn't be enough left to claim).

We have only used our account for meds and direct doctor/hospital bills which are easily verifiable and probably why in all of these years none of our claims has ever been audited/disputed. I really don't worry much about it.