Is mold remediation effective in the long term?
A good remediation company will do testing on both surface and air afterwards.  I would make sure there is a good amount of time past after cleaning to go through and do the tests - a couple weeks maybe. 

Your main concerns are basement and bathroom, so make them prove those areas too you.  If there is a lot of ongoing mold on a wall the it is likely it affected the surrounding areas.  So make them open up walls to prove there is nothing.  If they don't want to then I would expect the worst and know that first you might have more big issues that are hidden and two that you might need to do a lot of drywall and finishing touch work after you demo sheetrock in areas.  

Note: I would also try to put some of the risk onto the remediation company.  If I have to use one on flips we go over everything.  I get a price.  And then if they mess up they come out and fix the area for free.  For instance if they tell you that you don't have mold in a room but find it a month later they would come out and remove wall and fix area.  You have to get a reputable company and then you will usually only be given 3-6 months.  If you think the house was full of moisture/humidity for a long time then I would assume that most of my drywall has to be replaced.  For a 2000 sq ft house that is 10-15K.   Also know too, mold needs moisture.  If there is a little mold in a wall and it never gets more moisture, there is little to no harm...  I will say this, I rather deal with termites than big mold when buying a house.