Create an area within your community for people that use your competitor’s products.
Search for people on Google, Facebook and Twitter. Invite them to join and run this competitor division of your community. Don’t try to sell them, don’t spam their accounts or force them to do anything different.
You want your competitor’s customers to like your customers. You want that like to become trust. You want your competitor’s customers to envy the established community and want to become a bigger part of it. They want to participate in more activities, discussions and be treated the same as other members.
Side note: Any organisation that hasn’t started building a community amongst their customers might want to start now. Or you might do it for them.