Yes - or at least all the black?/red? fruited trees here are. I remember a purple blushed/white-fruited one in my youth that was growing alone... but there are wild trees everywhere here, and maybe it was being pollinated by the wild ones. That one had wonderful flavor, I wouldn't mind planting a small grove of those.
BTW, did I mention the roots? My line of mulberry trees runs on the back of my neighbor's 100-foot-long lot. When the lot was being excavating to build their house, they dug up those bright yellow mulberry roots running across the entire lot, all the way to the drainage culvert!

Some of those roots also ran all the way to my foundation, and shot up saplings next to the house. Those same roots ran under one of my gardens en route, and started to send annoyingly large roots up into the garden... unsurprisingly, the tree responsible for that was the first one I took down.
But I have to say, despite their weaknesses, I may end up actually planting some - in my back lot. They grow quickly & seem to be fairly tolerant of soggy ground, so they might be useful replacements for all of my dead white ash trees. Wish I could grow the white mulberry, but it can't be sold in Wisconsin.

Wish I could grow the very large-berried mulberries I've seen in the South too (saw some in Kansas that were HUGE) but unfortunately they are not rated for my hardiness zone. So I will likely cruise around when the berries are ripe, and seek a tree with higher-quality fruit to take cuttings.
While looking up Morus alba to see if Wisconsin classified it as invasive (it is) I found mention that some cultivars are monoecious. And that the dioecious males apparently expel pollen at high speed?!!!??