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It’s not a current feature of the plugin but theoretically may be possible by enabling the “Display Mode” setting. This would require that no visitors can log in to your WordPress. In that case, it would just be you (or whoever admin) posting chats for others to read. Note I haven’t tried this technique, just wanted to mention as maybe a possibility.
About server load, it really depends on everything going on in the stack. Should be fine though if it’s just one person sending out chats.
I hope this helps, let me know if I can provide any further information.