I love the in-app chat and ability to request services from the app, it is one of the best things about Marriott and something I missed when I stayed at a Hyatt.

Every hotel is different but typically you're talking direct to whoever is personing the front desk (although in some cases it may be the concierge). You will typically get an automated response acknowledging your message, and then a response from a hotel employee once they're available. I've found most hotels treat the priority of a message as lower than a phone call but higher than an email (i.e: a chat is something to be responded to as soon as someone is available, an email is whenever (if ever)).

As an almost daily user of the chat, my recommendation is to treat the messages like an email: ideally, the front desk employee should be able to read your message, action it and then close the chat without any further messages. If you want to have an extended conversation about something, then you should probably call or visit the front desk.

A caveat is that the Marriott app can sometimes be very delayed at delivering notifications, even by many hours in some cases. If I am expecting a reply (e.g: I have a question) then I will check the chat every few minutes just in case a response has been received but the app didn't notify me.