Thanks for the question.

Sawdust has been used both as mulch and being incorporated (amendment) into the soil for many decades.  However, the consequences of doing this differ between its use as a mulch or as a soil amendment.

Sawdust is essentially plant cellulose, which means that it contains high levels of carbon.  When this carbon decomposes due to bacterial or fungal action, large amounts of nitrogen are required.  If the sawdust is sitting on top of the soil as a mulch, especially if it is a thick layer, this depletion of nitrogen may have minimal effect upon the underlying soil and the plant's need for nitrogen.  If, however, the sawdust is actually raked into the soil the subsequent nitrogen depletion may have adverse effects upon the plant.  Consequently, when used as a soil amendment nitrogen rich fertilizer is often incorporated into the sawdust.

Another thing to keep in mind is that as the sawdust decomposes, whether as mulch or soil amendment, it will lower the pH of the underlying soil making it more acidic.  This would be desirable if one was growing plants needing acidic soil such as blueberries.  It would not be good if the surrounding plants needed a pH of 7 or more (less acidic).  In other words, before using sawdust one needs to do a bit of research regarding the pH requirements of their plants.

The following two sites will elaborate upon these points:

https://www.gardeningknowhow.com/garden-how-to/mulch/using-sawdust-as-mulch.html

https://homeguides.sfgate.com/mulch-sawdust-21473.html

You may want to bring these considerations to the attention of your neighbor