Most cases the messenger widget won't show because the domain it is on is not whitelisted. You can check from Settings > Inbox > Messenger > Whitelist section:

The embed script is not inserted correctly into the header or footer of the site of which you're installing the messenger on. Please follow the installations steps.

There is a conflict between our messenger script and some other script(s) installed on the same site. We recommend trying to switch scripts/plugins off to pinpoint the effecting script or plugin.