How can I share a pdf with another teammate I work with and make it editable for both of us?

Both of you can create an AdobeID for free without subscription. Your AdobeID consists of a primary email address that you setup in your Adobe Account profile, which conveniently  it also integrates Gmail, Facebook, and Apple ID's  should you prefer to sign in with this method.

However, in order to share and edit PDFs in a collaborating manner you both  must unlock the premium editing features of Adobe Acrobat in your Acrobat Reader mobile app. 

So yes, in order to edit and collaborate like you would with a desktop version of Acrobat Reader or Acrobat Pro you must subscribe (pay for a subscription of Adobe Acrobat Pro DC for example).

These editing features are somewhat limited in a mobile device when compared to the full desktop version of the Adobe Acrobat product. And it is worth noting that with a subscription to Acrobat Pro DC you'll have access to Adobe Acrobat online services, which provide a wider range of all the editing tools you may need to work with PDFs.

Last, another important observation, this Acrobat Reader Mobile app behaves differently, looks different and  provides different features when it is used in iOS mobile devices versus AndroidOS mobile devices.

This is something to think about if you want to minimize incompatibility issues between mobile device versions, feature limitations or integration with the other Adobe mobile apps (like Adobe Sign, Adobe Fill&Sign, Adobe Scan, Adobe Photshop Mix, etc), but most importantly to consider if you'll be able to perform troubleshooting of common  problems that can (and will arise)  in  a standardized and consistent  manner between two different mobile devices to minimize downtime.