I can't be certain, and most of this is going strictly by what I recall reading on other sites.

First, the AA was always based out of Owosso, while the GTW was based out of Durand.

They used to have their own lines between the two cities. The AA at one time actually had their own line from where Pitt Jct is (SE of Durand) all the way to Owosso. There was a diamond at Pitt Junction where the AA crossed to the north side of the GTW line. They ten ran parallel to each other until just before the station when the AA went to the north side of the station and the GTW line stayed south. They then each had a diamond over the Flint-Lansing mainline. If you look at Durand now, you can actually see where the AA mainline used to go on the north side of the yard. The tracks are still in place. This is the track that now comes off the main (Holly Sub) behind the fire department and runs on the far north side of the yard. This track is then separated at the west end of the yard by some trees. It now attaches to the line to Owosso just before Lansing Rd and I-69. Much of the AA ROW is overgrown now and impossible to see around Vernon. A few items are still in place including a bridge over the Shiawassee River. When the line turns and runs alongside M-71 you can see where the AA ROW was when driving on M-71. In Corunna I think they split, but it's almost impossible to tell on the overhead map view. When you get to the lead for San Yard it is easy to see where the AA came straight in. IIRC San Yard was either built after the split or was an AA Yard built to replace the Shops yard area. Once you got passed the east end of San Yard I think the customers were pretty much based on which line you were on. If you were on the southern line it was GTW, if you were on the northern line it was AA. You also had the Penn Central which run from Lansing to Saginaw and had a crossing of both companies east of Chipman St.

As was previously stated, the GTW did have trackage rights over the AA from Owosso to Ashley to access the line to Greenville.

What got interesting was when the GTW got the rights to all the customers in Owosso. IIRC the agreement that allowed the GTW to have access to the customers also included all customers on the GTW trackage rights section between Ashley and Owosso. There was some dispute about this and San Yard awhile back, such as who really owned it.