Every comment in the thread shows who wrote it and when. To add a new one, tap Comment at the bottom of the thread, type the text and confirm it — it’s added with your name and the current time, without affecting the ticket’s status.
Note — Assigning a ticket really does notify
When you assign (or reassign) a ticket to someone, that person gets the notice instantly: always in the bell, and also by email or WhatsApp if they have those enabled in their personal preferences — with a direct link to the ticket (manual 16 · Conversations, section 10). Assigning a ticket to yourself, or reassigning it to whoever already had it, never generates a notice — there’s nothing new to tell you.