Sorry if this was covered elsewhere and I couldn't find it! Maybe I'm wrong about how this
stuff works, but this was my experience:
Last week I was busy with life (work + wedding planning) and didn't login for a few days. A very nice person visited my street and left a note saying they had fed my piggies because their feeders were empty (I'm a bad Glitch piggy owner! I forgot about the piggies for days!). I received this note from my butler, since that is how the sender sent it. I wasn't expressly noting who the sender was since I assumed I would be able to verify it in my inbox later. But when I looked in my inbox, it wasn't there. It was gone! And trying to determine the sender from a long list of street visitors wasn't viable, nor did their message pop up in a spot like local chat for later reference. I really wanted to thank this person and feel like a jerk that I had no way to do so.
So, tl,dr: messages left with my butler should go to my mailbox, considering the butler functions as my mailbox. Messages left with my butler should not disappear after I click out of the message prompts.
Also, thank you Nice Person Who Fed My Piggies! I hope to one day return the favor.