Most phones show a little icon that looks like a passenger jet from above.
you won't really know if your other line is in airplane mode or not. You might get an indication if say your on iMessage or you're using RCS with Android and you have read receipts setup if you send a message to someone where normally you would get a receipt saying that the message has been received delivered or read and in this case you get no check mark or receipt it's possible they may be in airplane mode. However it is possible to be in airplane mode with Wi-Fi active and having the mobile data turned off. if they are in full airplane mode you would go straight to voicemail with a call get no read receipt. It is also possible that if they were on Wi-Fi calling they could receive messages and phone calls but being airplane mode as far as data is concerned. So the short answer is you don't really know unless you're looking at that lines phone or you're talking to that person with that line in person.
I didn't understand the question. How to tell if the other guy's phone is in Airplane Mode. Hmmm. Without their cooperation, I'm not sure if it's possible. I wonder if there's an app that will send out a notice when a phone changes mode?