Per AA's website T&C:
"Frequently asked questions associated with systemwide upgrades
. . .
Q: Can I share my systemwide upgrades with other people?
A: Yes, and you can travel with them, or they may also travel alone.
Q: How do I share a systemwide upgrade with another passenger if I'm not traveling?
A: Call your Elite Service Desk in advance to make the reservation and request the upgrade (the same way you do when you claim a mileage award from your account for someone else). Then the passenger can go to the airport to check in without you being present. If you do not call your Elite Service Desk in advance, the passenger cannot be accommodated at the airport unless you are also present.
. . . Upgrades are void if sold for cash or other considerations.
Members may designate another passenger to use their upgrades."
So yes, you can feel free to gift them as long as there is no quid-pro-quo, no trading, no sale, and no bartering (i.e no consideration except that the recipient's heartfelt thank you).
But if you'd love to gift them out of the goodness of your heart (or because you just can't fathom wasting a perfectly good SWU), you are welcome to do so. It will just take a call to the Elite Desk, as mentioned as above.
If they are still yearning for a home, feel free to PM.