First visual test forcomponents mounted properly and good soldering. Shiny joint is good,dull joint isa dry joint. Board should be properly cleand. There should be no residue of flux and solder on the board. Components vaues should be according to the specified values.
For electrical testing, i think that is not a job of assembler. PCB ASSEMBLY TECHNIQUES
Boundary Scan Testing, JTAG is best way to go, a good EMS should be able to do the complete job and provide you with fully working boards, no point in recieving duff assemblies.
In Circuit Test, Boundary Scan, Functional Test, Xvision test, optical inspection. You should ask your assembler about the tests they perform to ensure a correctly assembled pcb.