Do I need my Supernote device nearby or connected to sign in?

No. Sign-in only needs your Supernote account (email plus the emailed code) and an internet connection, because it goes through Supernote's cloud. Connecting to your physical device over your local network is a separate, optional step the app handles afterward. This optional step is only required if you want your notes to be attached to the task so you have them as reference later.