There was moisture damage to many walls and some ceilings as well.

I just rented out the unit I was living in, the next day the tenant below calls me that there is moisture in his ceiling above the bathroom.

@Travis Hargreaves those are big, sweet, gentle giants.

This isn't Canada specific, but I would be concerned with water that caused significant ground shifting, or accumulated in such a way that it could not evacuate the site and either posed damage to mechanical electrical systems, or caused mold, rot and other issues attributable to excess moisture. 

In general, you should do at least an annual walk-thru for condition.Tenants won't tell you stuff is wrong until it collapses and gets really expensive (like moisture invasion).

I am about to tackle the dreaded basement that has moisture.

The leak guy will have to trace those lines for moisture content in the dirt. 

As a property owner, my concern would be if the AC isn’t running in the summer to keep moisture out, would it allow mold to form?

If you install it over a wet subfloor or in a basement without addressing a moisture issue already present you may experience mold in the future without ever visually seeing an issue.

So the gentle call from the title company is a nice way for them to release the info, but also keep it quiet....especially if there is more assets in the trust than just this one piece of land.