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No, its not written to the blockchain.
If you use Keychain it will ask you the first time you write something to the blockchain on a new service.

Also HiveTasks will show you all your transactions to the blockchain. Chat doesn't generate a transaction.

Ahhh, okay, where are the BeeChat messages stored?

Posted via D.Buzz

On a private server that is owned by the beechat team. Do not consider those messages on beechat/peakd encrypted they should not be used for sensitive information. We have expressed a lot of interest to their team @aggroed and @reazuliqbal to find a way to have encryption of the messages.

Ahh, okay cool 😀👍