The grinding is the loud part, but what happens in the weeks afterwards is what decides whether anyone can tell where the stump used to be. It is a small amount of work and it is almost always worth doing properly.
Start with the chips. Grinding produces a surprising volume of them — considerably more than the stump appeared to contain, because the material fluffs up as it is cut. You have two reasonable options, and it is worth deciding before the crew arrives rather than after. Raked back into the hole, the chips fill the depression immediately and break down in place. Cleared away, they leave you a clean hole to backfill with soil, which is the better starting point if grass is going in soon.
If the chips stay, expect the ground to settle. Wood chips compact and decompose, and the spot that was level on the day will usually sink over the following months. This is normal and it is the single most common surprise people report. Plan to come back with a wheelbarrow of topsoil once or twice over the first year and bring it back up to grade.
If you want grass there, the shortcut is not to seed straight into a bed of wood chips. Chips tie up nitrogen as they break down, and seed sown into them tends to come up thin and yellow while the soil underneath sorts itself out. Either pull the bulk of the chips out and backfill with topsoil, or accept that the patch will take a season or two longer to match the lawn around it.
Planting a new tree in the same spot is the one thing worth pausing on. The ground is full of decomposing wood and old root material, and it is not the friendliest place to establish something new. If you have a metre or two of choice, plant beside the old spot rather than directly into it — the new tree will get going faster and you avoid inheriting whatever took the last one down.
For a bed rather than a lawn, the chips are less of a problem and can even be useful as mulch on the surface once the depression is filled. Just keep them on top of the soil rather than mixed through it, and keep them off the stems of whatever you plant.
Planning what goes back into that corner of the yard? Call Alexander's Stump Grinding & Tree Service at (470) 939-8929 for a free estimate, and tell us what you have in mind for the spot — it changes how we leave it.