I think most people are misreading the OP.
From what I’ve seen, the “Live” system is shared between the two platforms, so you can ‘friend’ each other, send messages and such. However, they are fundamentally different platforms, you cannot play an Xbox 360 game on the Xbox One (and hopefully obviously, the 360 cannot play One games either, otherwise you wouldn’t need to buy a new console). So the short answer to “can they play together” is “no.”
I guess theoretically if a game came out on both platforms then it may be possible to make them talk to each other cross platform, but that would be down to the individual game developers to implement / support that functionality, and that’s probably going to happen as close to “never” as to make no odds.
On my 360 it shows what friends are up to, what games they’re playing and so on; those who’ve upgraded to Ones just show up as “Xbox One” and that’s all you get.