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You can leave a private message conversation and choose not to receive any further replies from that user or group. There's no option to delete.

Leaving the conversation will remove it from your direct messages list.
 
Hi,

Can someone tell me how to delete private messages here? Thanks!
You can do what sman789 mentioned, but if you have a private message you sent, you can just edit it out. Just edit and remove the text. Its like any other post in the forum in that sense, except just between members
 
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