Yes, you could do that. To do this, go to Settings>Mail,Contacts,Calendars>Add Account>iCloud, enter a different Apple ID to create the account and turn Notes on. Then do the same on the other device. You would then need to go to Settings>Notes>Default Account and set the new iCloud account as your default, and open the Notes app, tap Accounts, and tap this account to be sure you were creating new notes in the shared account.
If you have existing notes that you want to get in this account, you'll have to open each one and copy and paste the text into new notes created in the shared account.
Edit: on the second device, go to Settings>Mail,Contacts,Calendars>Add Account>iCloud and sign in with the shared iCloud account ID you created on the first device (rather than entering another new ID, which would create a forth account).
Message was edited by: randers4