I see a mod posted a sizable bounty to my question:
I'm wondering if I should be accepting an answer in order for the user who posts the correct answer to be awarded for the bounty or if the person issuing the bounty should do it?
Clicking the green checkmark does not award the bounty to a user. As far as I am aware, the only way a bounty is awarded is if the person who started the bounty clicks the button to award the bounty. Otherwise, half of the bounty will go to the highest upvoted answer at the end of the bounty period, if there is an answer with at least a score of 2.
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your own answer.)
If you do not award your bounty within 7 days (plus the grace period), the highest voted answer created after the bounty started with a minimum score of 2 will be awarded half the bounty amount. If two or more eligible answers have the same score (i.e., their scores are tied), the oldest answer is awarded the bounty. If there's no answer meeting those criteria, the bounty is not awarded to anyone.
If the bounty was started by the question owner, and the question owner accepts an answer posted during the bounty period, and the bounty expires without an explicit award then we assume the bounty owner liked the answer they accepted and award it the full bounty amount at the time of bounty expiration.
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.