Your Twitter app will have a consumer key and secret. When either of you authenticates with the app, you will have your own access token and secret based on your own Twitter account. Your combination of consumer key + secret + access token + secret will be unique to you, and your collaborator’s will be unique to them.
There’s no support for using the teams feature in Tweetdeck to enable shared access to applications in the API. You’ll need to separately authenticate to the app as either of the accounts in question.