Overwatch doesn't currently offer players the opportunity to play with players on other platforms, but it may be possible in the future.
Responding to a fan query on Twitter, the Overwatch account said that the team will "keep an eye on cross-platform possibilities on the consoles." Such a feature would be restricted to consoles though.
It added that it has no plans to allow PC players to play against console players. "Mixing mouse/keyboard and controller can have balance and competitive issues," it explained.
Rocket League recently made it possible for PlayStation 4 and Xbox One owners to play against one another. We don't have a timeline as to when Overwatch will embrace this option, but it's reassuring to see that platforms are breaking down barriers to playing with friends.