I'll throw in one opinion of mine here. I work for a private semi-rural service, that really only serves one hospital. We perform (medics and Intermediates only) field blood draws, for use in the ER. That particular ER (and labs, i presume) like us to use a particular brand vacutainers, and our administrators dont really like buying them...
So we use supplies taken from the ER (with permission). When we would normally start a line (or lab draws are warranted), we use their brand vacutainers for the collection. We restock whatever we use (draw 3 tubes, take 3 tubes), but have a small supply back at our garage.
It is all agreed upon-- and since when we start a line we bill a flat fee (for the saline lock, not the labs), and it saves the ER some work, and they can bill for the lab test (I presume, including supplies), it is a win-win.
In terms of linen, it is usually one 1:1 basis. The same hospital from above has special EMS linen (different color). There is a set ammount of it, and when the supplies run out, they put out a memo to services, requesting it's return. I presume the system works out... but it is mainly an honor system.
The other service I work for is much more lenient. The hospitals we serve dont really care how much linen we take-- as I guess it all makes it back to them eventually. Fairly often, on our last discharge of the day, we leave the linen from the stretcher with the patient at home. Our supervisors dont like us taking dirty linen back to the garage at the end of the day, and it is often inconvenient for us to return to the hospital.
Again, we really dont hear anything about it, so I dont think it is much of a problem...