From what I've read, they'll cross. However, I wonder how much of an issue it would be... Don't they grow via root, more than reproducing via seed? If it's more by root, then I wouldn't worry about it too much, but if by seed, I'd put a patch of each in different areas.
Any crossing would affect the seed. The berry - which is what you're after - bears seeds, but is not itself the seed. It's the fruit, outcome of the flower.
Shouldn't be any trouble.
And, yes, in people's gardens the proliferation of plants is a matter of new shoots coming up from the roots, and developing at a bit of distance from the mother plant. You can dig these young plants up, cutting them off from the old root system, and plant them into the rows of your patch.
The only issue I've ever read about with raspberries is to keep red and black raspberries away from each other because of disease--but I don't remember the specifics, sorry
My Mom grew up on a farm in BC's Lower Mainland (Chilliwack) and they had acres and acres of raspberries plus every property there is edged with huge blackberry bushes--and they sold their raspberries to Fraser Vale Foods, so if they were good enough for the packing plant, I'm sure it's fine to grow the two together at home