For compressors, in most cases I've come across, no load string test is done. If you want full load string test, you should make that clear in your RFQ. String test is not mandatory, but it does clear up teething problems that the vendor's comm engineers may be ill equipped to deal with when the unit is installed at your plant.
@OP: It all comes down to where you want to find any problems. NONE of the testing is a requirement - unless your bid document makes it so. That being said, there are certain test criteria that are strongly recommended to enhance both safety and performance.
Personally - 1) do the factory acceptance testing of individual components (motors, generators, compressors, pumps, etc.). Follow up with 2) perform no-load string testing for major (or critical) processes, usually at the driven-load supplier's facility. For really critical stuff, add in 3) a loaded string test where possible. And finally, 4) actual loaded string testing as part of the on-site commissioning process.
Why do 4) if you've already passed 3)? Because the situation at site is a combination of a lot of things beyond your immediate control - and 3) was based on what SHOULD be happening at site, not what is ACTUALLY happening.
Converting energy to motion for more than half a century
For gas turbine generators, problems appear with sychronising the new machine with existing parallel running machines which are not resolved easily - there is nothing you can do about this at the OEM site.
Similar problems appear with getting a new compressor to take up load / deal with capacity changes through the new machine's capacity control / antisurge control system since it involves interaction with the existing master capacity controller.