Hi! Glad you're considering these because they are probably one of the coolest "not-quite-Connection/modern" Tamagotchi "but still kind of does the connection part, in a way," in my opinion.
They're a pretty classic, must-have for collectors. They're cute, pretty basic, not too demanding.
I've never tried running one without the other, but I imagine you can't "advance" to the next "generation" (just to be clear, there is a limited amount of times you can move on to the next generation, unlike on the modern Connections, where you can do that for pretty much as long as you want) and get new characters. So in sum, I would say yes, to get the full experience, it's recommended that you run them both simultaneously.
I'm actually running my Osu/Mesu pair in my log right now. It's been a while; haven't run them since Valentine's day nearly 2 years ago, but if you're interested in a taking a look, I logged about them
here. I only went for one generation because I was participating in a hatch at the time, and one of the stipulations was to run one generation...which was great because that's about all I had time for anyway.
You're probably interested in some more detail, right? Unfortunately, I was always too busy to do super-detailed accounts but there are a few other resources I would recommend:
TamaTalk Osutchi and Mesutchi Chart (plus some other basic questions)
Our
Pixelmood Osu/Mesu page
Mimitchi.com's specific
page on the Osu/Mesu
Hope this helps a bit. If you have any more specific questions, feel free to post them here.