As had already been stated, some people view all non native species as potentially invasive and this is probably a valid concern. Sawtooths have been in the U.S. since the 1860s though. My dad and his neighbor have probably 50 sawtooths with the neighbor's trees being over 20 years old. Like catscratch, I have never seen a volunteer survive. That being said, I have read some people say that sawtooths are invasive in Illinois and also Alabama. Some people have stated that sawtooth acorns are not as good for wildlife as native acorns. This
article states that sawtooth acorns are not as nutritious as native oaks, but the authors compared apples to oranges in providing the nutritional content without standardizing the moisture content of all acorns within their table which affects the nutrient content of the acorns.
If you are looking for fast growth and early acorn production, sawtooths may work for you. There are also native oak species and oak hybrids that grow fast and can produce acorns in less than 10 years, so they may need to be considered also. Drop time also needs to be considered. Where I am in south Mississippi and Louisiana, the sawtooths drop in mid to late August. They usually only drop for a couple of weeks and then they are done. It looks like the drop time is delayed the further north you get, so your latitude and drop time may play a role in your decision. Most native oaks tend to drop later and for a more extended period of time.