Private onsen in Aso Onsen
A bath you have to yourself: no other bathers, and usually no tattoo problem either.
Is there a mixed onsen in Aso Onsen?
Yes: Jigoku Onsen Seifuso. If what you want is to bathe together as a couple or a family, a private onsen is usually how it is done in Japan: the 1 below has a bath your group uses alone. Where Japan’s mixed baths are, and what to wear