There's never a clean way to with any Gardening tasks, even watering anounts to me getting hands and other areas dirty or make a mess due to one simple job leading onto another LOL.
I think Doug has given great advice and would be doing the same myself it I was gathering seeds, IF weather is dry, a paper bag is also enough so long as moisture cant dampen the seeds before planting,
The trouble of bringing the seed heads or flowers indoors to ripen at this time of year is, they are then being offered a much warmer environment and most likely less air flow, but you will find all of us garden people do the same type of stuff but adapt it to suit our needs, however, there are simple methods of doing some work that you really take a big chance with end results IF you vary the method too far from the basics.
Is there any good reason why you cant weed out all the self sewn seedlings BUT, save a few to grow on as new garden plants by potting them up when large enough to handle and then transfer them to where you want them to grow next summer, they usually make far hardier transplants if done naturally by themselves, just my own opinion.
Good luck, hope all is a success.
best Regards. WeeNel.
You might want to use scissors when harvesting the dry seed from the vines. That way, you might eliminate shaking so many seed off onto the ground before saving. Most morning glories produces scoodles of seed. After I save wanted seed, I'm thinking of trying to burn the residue from my 2013 morning glory plant in hopes of eliminating a lot of weeding out next year. Mine is in an area that I can have a small burn.
I will continue to do what I have done in the past, I usually cut down the lines that they are hanging on, and put them in a plastic container and allow them to dry in our very dry basement... may take weeks. Abit of a mess to go thru the vines mid winter and gather all the seeds. Even using this method, there are some seeds that have already dropped to the ground and germinate next year.