All the Sears stores near me are in large shopping malls. I doubt they're anything more than tenants, although I have seen some malls where some of the major stores own the mall in a partnership.
The two big traditional Sears locations in the Bay Area were in Oakland and San Francisco. I'm pretty sure that Sears owned them, but after both were closed in the 90s, they stayed unoccupied for years. The one in Oakland became a loft apartment complex. The biggest tenant was Jerry Brown while he was Mayor of Oakland. The store in San Francisco is now a Target. I looked up the history, and Sears sold it earlier with a leaseback, so they were playing the real estate game for years. The cost of needed repairs and a general lack of business is what killed that store.
I remember when we needed something, we would go to Sears. The details about the deal are that Sears stills gets to sell it for 15 years without having to pay royalties. I'm assuming that means they get it at cost, unlike other retailers. Still, as a near exclusive brand, Craftsman would drive customers into Sears stores. Not sure what was up with the Ace deal. The last Craftsman tool I bought was a bottle opener. It looks almost like a wrench.