BWT, I might be imagining this, but didn't, at first, the O's OR the Nats be at home? I could have sworn that at first that was part of the agreement, that they couldn't have home games on the same date. Somewhere along the line that changed though.
They generally speaking play at home at opposite times just like the NY/LA/ Bay Area teams. The first year of the Nats, I don't think it was too uncommon since they were playing with the Montrael schedule. I'm guessing they have about 10-15 home games overlap per year, and it's usually a single day not an entire series. Example: Nats end a homestand on Monday, O's begin one on Monday. This is one of those rare ones, where they overlap the entire series (Nats were off Monday).
A lot of the venders such as Clancy and Howard work at both parks, so it's nice for them to not overlap so much. I'm not sure how much, it takes away from their respective audiences when they overlap. I'd imagine very minimal, I don't know how many people (not corporations) have season ticket packages to both but I'm guessing not many. When the Nats first came, I think many of their fans were still following the O's on a more active basis, when they moved to Nats Park, I believe that changed. At this point, I think you have a primary rooting interest.