TLDR: The Thirdfort check is completed by one individual per check, with each person needing their own verification. In the app, checks are not assigned to a specific person until their name is submitted, so multiple invitees on the same number can complete any check.
Who needs to complete the Thirdfort check?
In your initial invite to Thirdfort, this will confirm the name of the individual the check is for. Each check is designed to be completed by one person. When multiple individuals need to be verified as part of your transaction, they will each need a check.
Which check in the app is mine?
When completing a Thirdfort check, this is not specific to you until your name has been submitted. This means that if you and your partner are invited to complete a check on the same number, either of you can do either check.