Chat widget guests will receive back an automated note, "Message not received." so that they understand their message did not reach you. Your chat widget would also have looked offline in that case. Years of experience with widget chats has shown us that chat widget guests will almost never be there to receive a reply from you upon your return, which is why we don't try to hold and deliver these messages later.
Of course, this is why we provide a presence API to inform your guests that you are not available and to instead display alternate contact information or maybe an email form. That "we're offline" appearance is under your full control and you can literally make it appear however you'd like when you're not there.
For texting guests -- because they have a real identity -- we save the offline message and display it for you the next time you log in, so that you can deal with it at that time. We also tell you how long ago the text message was sent. Don't worry, we also send a customizable offline reply to the guest to let them know you're not there.