When I sell on Marketplace using my fake account, I just keep my phone and/or iPad near me and hear a ding when I have a new message. I use the Facebook app for my fake account for this reason.
I use browsers for my real account. I don't see or get notices from it, because it's not an app. Once in a great while, I'll get a random email that someone has messaged me in my real account, since my "real" email is connected to my real FB account. They are few and far between and hours after someone has messaged, and I have usually already seen it, so I have no clue what that's about.
Also, if possible, you're much safer meeting someone in a public area than at your home. People who I meet to sell my items have no idea what my real name is or where I live. I meet in front of Petsmart, which borders a busy road. Not trying to be bossy, but I've sold on Marketplace with little issue for several years, and I just want you to be safe.