Manually awarded bounty:
The accept (you) could be, applied when an acceptable answer existed at any time, without effecting the bounty , before , during or after, or even never :-O, that does not affect the 3rd party bounty. If the 3rd party bounty user awards manually
If I had bountied your question, I would want the original person asking the question to Accept the answer that does answer thier question, or solve thier problem, and not accept anything that did not answer the question.
I have been there before when trying to figure out where to hand off the bounty, before it expires, it would have helped if the original question was answered enough to be accepted. All that would be the thing to do anyways.
So while you wont be handing out a 3rd parties bounty (as already pointed out) if you ever actually got an answer an accept or even commented as to the uselessness or helpfullness of the answers that arrive, it could come in handy :-) to the person placing the bounty.
A 3rd party person putting up the bounty, may also have thier Own (varied) adjenda, as displayed in the text of the bounty. Putting up the bounty on a question to get specific information. They do not create a new question because it may be marked as a Dupe. So there could be more going on, with 2 slightly differing ideas, 2 completly different awards for good answers. (the bounty and the accept)
Auto Awarded Bounty:
An accept on the answer during a bounty, will get the bounty total, if the bounty time expires (and 24 hour leeway), and the bounty placer does not pick an answer to award the bounty to.
With no accept , if the bounty expires without the 3rd party awarding the bounty, the voting that you or anyone else is doing can still effect where 1/2 of it is then dumped. 24hours after the bounty ends if the 3rd party has not specifically put the bounty somewhere.
While you cannot get your Own bounty back that you created, "You cannot award a bounty to your own answer." A 3rd party bounty could be given out to another persons (you) great self-answer. So solve those problems.