Last I heard (which was a couple of months ago) it hadn't been decided yet. All (or most) of the Shenfield Metro stations are pretty much shoo-ins, although likely everyone wants Stratford. Hayes and Harlington eastwards (with the possible exception of Ealing Broadway) are pretty likely I'd have thought, Heathrow almost certainly will remain BAA, and FGW will likely want to keep their hands on Slough. Other than that I'd say it's up for grabs.
The essential things in life are seen not with the eyes, but with the heart. --Antoine de St. Exupery
Note - speculation involved! I think it is pretty clear that stations east of Stratford to Shenfield will move to the Crossrail concessionnaire. I would also expect that Canary Wharf, Custom House, Woolwich (if fitted out) to be controlled by the concessionaire as these "stand alone". Abbey Wood is unclear and as already said will probably be linked to however the rail franchise devolution issue is resolved. In the Central area I believe that Whitechapel, Liverpool Street / Moorgate, Barbican / Farringdon, Tottenham Court Rd and Bond Street will be LU managed and operated.
Paddington Crossrail is largely physically separate from the rest of the LU station so I suspect it may be managed separately by Crossrail. Looking at the planning brief there are link corridors to the Bakerloo and Circle / District Lines but it is hard to tell whether these are paid side or unpaid side links. This would be a factor in understanding where management boundaries are put.
I have no clue what is happening west of Paddington as I don't know the extent to which Crossrail "takes over" services in totality and how much will be shared with FGW (or successor TOC). I know Crossrail take over the Heathrow Connect service before the main route opens.