I was recently invited to a PM conversation involving a moderator and member of my site.
This turned into quite an important conversation because the member was discussing paying for advertising on my forum (and had just paid, and wanted to know if I’d received the payment).
Unfortunately I never saw her most recent PM message. Discourse never alerted me to it in any way. I only found it when I saw an unexpected payment in my bank account from that member! Eventually I browsed to my messages page, and there was the message.
IMO when I’m invited into a PM chain, I should be notified of new messages in that chain.