Not necessarily.

Not sure about 135, but in 121 the airlines can hire and train you prior to all background checks being complete as long they're done before you actually fly a plane (IOE for 121).

Most airlines complete the checks before offering a class date. They do that to avoid an awkward situtaion where have to they fire someone who just quit their old job, possibly giving up many years of seniority/longevity. That could be result in a lawsuit if the background issue is not a SIDA no-go, but rather something the company just prefers to not tolerate.

Some lower-cost operators (notably regionals) will not spend the money in advance, since people no-show their classes all the time. Every new-hire class I've done prior to the majors had at least one person pulled out of class in the first week or two and sent packing.