To chat to with another user you firstly need to be ‘matched’ - this means both users need to swipe right on each other’s profile. Once you’re matched, you can start chatting.
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Chatting is completely a FREE feature for everyone. 

However, if you wish to skip the matching process, you may send a paid Chattylike to someone to start a conversation. This works like swiping right but with a feature to attach your text message.

To use the Chattylike feature, just tap the smaller like icon on the Swipey screen.