You might want to browse through the
FAQ. You don't have to use the cloud storage feature at all, and can keep everything on your local network, and with the team package, you can even block access to the cloud storage.
I don't believe that the admin requires a subscription, but I've not verified that. It would be odd, given that many places large enough to use the team version would have an IT guy who isn't an end-user of the software. But I wouldn't put it past Adobe.
If you want to use the software simultaneously with the other user, you'd need two seats. If you only want to use it once in a while, you might be able to get by with simply transferring seats back and forth using the admin console.
From the point of simplicity, sharing credentials is probably the simplest way to go, but from a business perspective, it opens you up to some risk. For example, your user will be able to do account management things, like look at credit card details, or sign you up for more products. You're also breaking their terms of service agreement, which says that as an administrator for the business, you can create/manage Adobe IDs, but any other use is not permitted.
For the extra $20/month, that's the way I'd go.
edit: with the team version, I meanAlthough, at the moment, they're running a limited-time offer to lock in for 2-years at $40/month/user, which is cheaper than purchasing a solo membership.