How to tell if terminal session is running within screen

Typically, $STY will be set.

So:

if [ -n "$STY" ]; then
  echo "I'm most likely running under screen"
fi

$STY is typically what you need to talk to your screen. That is used to construct the path of the Unix domain socket used to control screen (something like /var/run/screen/S-$USER/$STY.

Now, that won't work for instance if from that screen, you ssh to another machine. The shell started there won't have $STY in its environment, and that wouldn't be of any use to it anyway, since it wouldn't be able to access the Unix domain sockets on your machine.

However, it is possible to query the terminal with an escape sequence to determine its type:

if [ -t 1 ] && [ -t 0 ]; then
  s=$(stty -g)
  stty -icanon -echo min 0 time 3
  printf '\033[>c'
  type=$(dd count=1 2> /dev/null)
  stty "$s"
  case $type in
    (*'>83;'*) echo "this is screen"
  esac
fi

Another approach, as suggested by @val0x00ff is to check the value of the $TERM environment variable. That value is meant to tell applications what type of terminal they're talking to. It is set by screen to something that starts with screen (as screen can implement different variants of terminals). $TERM is passed accross rlogin, rsh, telnet, ssh. It's not as guaranteed to work as the above but is simpler and less intrusive.

case $TERM in
  (screen*) echo "I'm more than likely running in screen"
esac

Like for the previous solution, you may be running in screen, but you may not be able to issue commands to it with screen -X for instance. However note, that you can pass commands using escape sequences (though it's not enabled by default for security reasons).

When running in screen your $TERM environment variable changes to screen. You can check in the script e.g

if [[ "$TERM" == screen* ]]; then
  echo "Running in screen"
else 
  echo "Outside screen"
fi 

That would be one of the approaches. echo $TERM outside the screen and again echo $TERM inside the screen and see the difference