Im confused by what you are saying. If the council offload the Shay to a 3rd party(There are no other interested parties BTW) then neither team would be running the stadium. Again if its a 3rd party running it the both of the clubs would be working in partnership with that 3rd party rather than each other.
I can't see anyone buying the Shay, you have nearly a 1m outlay just to get the pitch so it is fit for purpose and the reality is that the 2 clubs combined only bring just over a couple of thousand fans in each week. There is more a chance that the longterm outcome will be the land is sold and both teams make alternative arrangements as to where they play.