Make sure your Facebook Page Visibility is set to “Page Published”
If it is set to “Page unpublished” then your Facebook Page, Messenger only Visible to the user who has a role in that page.
If unpublished another user can’t send the message from Messenger, so the messenger icon in the web page may not appear for users who are unable to send the message or for logged out users
One of the easy to check your messenger is accessible to other user is login to Facebook from another account.
If unable to open that messenger account, or the message is not sent – can understand that page visibility may set to unpublish or messenger may be blocked.
Publish Facebook Page
From the page,
click on Settings,
At General tab
click on Edit at “Page Visibility“
select “Page Published” and “Save Changes“
Clear Country, Age Restrictions
If country, age restrictions added to a Facebook page, that page messenger may visible only to the users who logged to Facebook in that browser.