Most washing machine soaps/detergents are high in Sodium. This causes clay soils to "gel". The clay gets sticky and the soil loses aeration. If you live in an area with high clay content, really large quantities of gypsum helps counteract the detergent issue. Gypsum is CaSO4-- and the Ca replaces the Na (Sodium) in the soil gel, allowing the Na to wash down and out of the root zone. In high drought areas with "adobe" soils, the sodium can build up and be negative (salts the plants).
If you live in an area with sand soil - high drainage, this is not an issue. If you live in a hard water area with lots of replaceable Ca or Mg, the sodium is taken care of by the water.
In former times (and some dishwashing detergents still), detergent had sodium tri-phosphates as the anionic. This provided substantial phosphate fertilizer to the plants (and caused lake waters to have algae blooms from the fertilization). Phosphate detergents are widely banned for this reason.
As long as you are not watering uncooked vegetables, you can consider grey waters generally hazard free in terms of parasites/microbes. Most success installs have some sort of sump (bark, horse bedding, etc) so water does not appear on the surface. I've installed a couple of pump assisted system and I cannot recommend those (maintenance nightmare, and expensive was to waste a grinder pump)>