Yes, it can be done and you have several choices as to how to do it. Seeds are the least reliable and slower way but I will list it here anyway. This information is free, it is the result of many years of trials and errors. I include it so you can propagate paulownias at your own pace.
If you have a paulownia elongata tree in your back yard and you want to use its seeds to propagate it, then you have to wait until the tree decides to make some seeds. First thing is the flowers, from them the pods are made. Elongatas are funny about this, they may flower on the first year like they may flower 10 years later. After the flowers come the pods that are about 1.5 inches in diameter, then you must wait about a year until they mature.
When the pods turn brown, take one of them down, squeeze it until it opens and if you see about 500 seeds fly all over the place, then the pods are ready and at this time you must cut the tree down to get your pods. Don't worry, the tree will start growing back up immediately, in fact it will make many branches. Allow only one to grow and support it, so the wind does not take it down.
What to do with the seeds? You can plant them. Do not cover them at all, they require light to germinate. The ones you don't plant, save in a sealed bag in the fridge at 40 degrees. Once you have planted the seeds, keep the soil moist, not wet, within 8 days the little triangular leaves will start to show. Carefully remove the little plants and plant them in individual containers. The following 2 months are crucial, mortaility at this stage is very high and you may loose a very high percentage of trees until the plants are about 2" tall.
Mortality is very low now and most of your trees will make it. Transfer to a larger container so they can grow.
Propagating by root cuttings.. Get your little shovel and start digging straight down about 6" away from the trunk of the tree. The larger the tree is, the larger the roots will be. You may want to do some of the digging using your hands, the roots will be about 10" down and you don't want to damage any of them. Once you find a root, very carefully start digging farther away from the tree, following the root to expand your find. DO NOT PULL on the root, you will only remove the skin and that makes the root unusable. They are very fragile. Continue digging until you can see about 6 inches of the root or more. 1" roots are not rare, those can be cut in 2" lengths.
Get your cutters and cut off as much as you can see off the tree. Now you can cut it again to make 3 inch cuttings for thin roots. Dip them in a rooting hormone and plant them either in the ground or in a container about an inch deep. Water them lightly once (ONLY ONCE) and leave them alone. What will happen if you water them more than once? The root will become mush and die. The roots will produce a shoot within the next 3 weeks. Repeat the same digging process all around the tree and make sure you put back all the soil so the tree continues to grow. This process willo not hurt the tree, it will make new roots to replace the ones you remove.
Grow a tree in a 5 gallon container until it is about 4 ft. tall. Cut the top off leaving only about 6". Turn the container upside down and carefully remove the tree. It will have plenty of roots, some of them 1/4" in diameter or larger. Take those off and save. Put the tree back in the container along with all the soil. The roots? You already know how to use them. I have gotten trees to grow from 1/8" diameter roots.
I like this method the best. This is something I did last year. In October I cut off a 20 ft. tree that was about 8" in diameter at the base. It was cut about 10" off the ground. Early Spring I started seeing little shoots showing around the tree. There were at least 30 of them. By the way, this is called coppicing. I started digging around the tree looking for the place where the shoots originated. They were coming from the very base of the tree about 8" below the surface. Since the shoots were underground and were not in the light, they were mostly white. I started cutting them off the tree with a scalpel trying to get as close to the base of the tree as possible. I then took those shoots, and after dipping them in rooting hormone put them in individual containers with the tops sticking up, and watered them once. The little shoots did not miss a beat, the leaves did not even droop, they just continued growing as if they were still attached to the trunk.
2 weeks later, I again had a lot of new shoots. You guessed it, I got at least 100 trees from that one trunk. It will be doing the same again this year.
Now you know how to do it, go get them!