Can I check or see if the recipient has read the message I sent them?
Where in Facebook do I go to see if this is possible?
From now on, yes. Facebook provides Read Receipts for its users.
How do I know if a friend has seen a message I sent?
When someone sees your most recent message, it will be marked as seen. That way, you always know who got the message, and who didn't.
Messages are marked as seen if the person is actively chatting or checking their messages.
This feature is part of every message you send, whether it's from your phone or the web.
No, this is not possible. FB messages are not even email, and even with email nowadays return receipts are not really widespread outside of corporate mania.
Yes, you now can (Facebook Help Center link). Just go to your message inbox; you should see the notification at the bottom of your message.
Related Web Apps SE questions:
How to disable "Seen by" in Facebook chat
Does "previewing" a Facebook message mark it as Seen?
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