Nandina domestica is not one of those plants that I like to propagate. It's a nice enough plant but I've always been of the opinion that it's overplanted in way too many landscapes. Every commercial business around has at least one and when a plant is that common I tend to cringe when I see it. Most of those are the dwarf varieties with the light green colored leaves that tinge a little on the red side. If I had to choose a nandina (and I wouldn't) I would go with the full sized Nandina domestica with dark green foliage and red berries that grows 6-8 feet tall. Ironically that's just what we have! Our 'Heavenly Bamboo', as it's called (don't be mistaken it isn't really bamboo), is in our front garden and is completely loaded with bright red berries during the winter. I don't really need anymore nandinas so I don't try to propagate these plants but there are two ways how they are propagated here.
How to Propagate Nandina:
One way to propagate them is through the seeds. Inside the berries are seeds that easily germinate in the late spring or summer if allowed to grow where they drop assuming that the birds don't get them first. There are multiple plants as I type beginning to plan their invasion of my front garden. They are small now but will quickly grow to overtake that small garden. Once germinated they can be moved before they get too large.
The second way I get more than I need of nandinas is through their runners. The mother plant frequently sends out roots that begin to sprout their own stems just a few feet away from the parent plant. If you dig into the soil and find the roots that attach the "baby" nandina from "mama" nandina and sever them with pruners you can easily gain an extra plant. There are several nandinas that I need to prune out of that garden as well.
If you want a third method to propagate nandinas you could try root cuttings or stem cuttings during the warmer months.
It's also very important to note that nandina is invasive in many states including here in TN. There may be some cultivars that are less dangerous than others but I highly recommend not propagating nandina or even planting it in your garden. I've only written this post in response to a Google search which also gives me an opportunity to discuss it's invasive properties. Our nandina came with the house and while we enjoy it's beauty (and it's ability to save my lawnmower) I'm avoiding intentionally propagating more. The problem is it just does! Which is why it's listed as an invasive plant. Native plant alternatives from the Tennessee Exotic Pest Plant Council list: Leucothoe axillaris, L. fontanesiana, Xanthorhiza simplicissima, Agarista populifolia, Hypericum frondosum, H. prolificum.
I've read accounts where nandina is perfectly well behaved so you make your own choice, but invasive or not it's way too overplanted for me to add more to my landscape!