As a clan, you should be unified but if there's a charter leader, they can take business to private or in topic. It's been going on here like this for the longest time.. just check out Gangs Chat, numerous gangs there are multi-platform, it doesn't interrupt or cause any issues.
All you have to do is share in each others conversations, not a hard concept to grasp. Post snapmatics, talk about events coming up, create order. If Xbox has some news coming up then they can make a
"XBOX Members Attention:: (insert news or updates here)" same goes for PS3 side, they'll read it and continue about regular discussion.