I have known of issues with "private owner" locos being at sites and then due to corrosion exposing asbestos which then became an issue for the site owners, and required some cajoling to get this removed with the threat that the loco would have to leave if this wasn't done. I don't know how easy that is to do if say work has commenced and means a loco must be moved by crane rather than shunted.
Personally, if I ran/owned my own site I would be very careful with any private owner locos/stock, unless there was evidence of funding available for the work, or the owners had a good track record of restoration, maintenance and site etiquette. Just having conditions for working on site including basic HSE compliance may put some people off working there, and they forget that although it's their engine, legallly the railway that are working at has liability for their safety.
But if items are to be stored awaiting restoration with active fund raising ongoing, then a tin of paint every so often does go a long way to stopping the rot on open wagons, and a tarpaulin (or two) can prevent further ingress of water to carriages or engines. I admit, this does all involve some money, but wire brushing and generally cleaning up is free, so as long as there isn't access to do it, then there is no excuse for not doing it.
My personal bug bear is storing rust hulks on the second line which means that a) the public are exposed to it every trip and b) it prevents work as you are now on the running line, which causes a clash at say weekends when the owner says may have time to get out, but can't get access. That does just exacerbate the issue, and eventually these items will be beyond saving.
Of course if money was no object I'd build a new site with either a US style round house with individual sheds for overhauls, or a Crewe style system with a traverser, with each side of the traverser leading to an open area and then a shed, which would be secure and lockable with metered supplies. Locos etc. could be overhauled, brought out for startup/steaming etc. and rents payable along with services used.
I feel however it may need a big site, and somewhere to store items that the owner's default payments on... but it would clear up a lot of the heritage sites, which in turn may increase demand for engines, which may mean some of these projects do actually get completed.
Or of course my fantasy site just gets filled up with rusty hulks that the owners walk away from and I need to work out how to get a 9F (other engines are sadly available all too readily) with no wheels
out of a shed.....