Models and string types

I have a P-bass and a J-bass, both with roundwound strings. I want to get a set of flatwounds, but I'm not sure which one to put them on, or to put flats on both of them. I've heard people have flats on their P-bass and rounds on their J-bass and some the other way around. What do you all recommend?
If it helps, besides jazz, I play mostly rock and funk, like The Beatles, RHCP, Fleetwood Mac, Tower of Power, James Brown, etc.
 
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I recently put the TI Flats that I had in my P on my J bass, it has Seymour Duncan Antiquity II pups. I love it.

I didn't like them on my P, they were too bright, I always had to turn the tone knob down. I put GHS flats on the Precision,they sound better for the P, in my opinion. It's a Midnight Special.
 
I’ve had both on both, and enjoyed it all! Final decisions ultimately came down to which string fits which bass.

So my advice, Decide which of your two basses sounds and plays the best, and try flats with the other one. Go from there.