Check the helical "cut" on the shaft for a burr etc.
A little silicon type spray lube may also do the trick if you have access.

Also note the word "little" regarding lube for the shaft.Usually the difficulty is that the gear is not traveling down the shaft. Hence why Bill noted to check for a burr. At times the pinion itself is worn or the drive disc inside is glazed. Many of these pinions have a rubber disc above the metal drive disc at the bottom of the gear. Lube there and you will be dead in the water, er snow. A very light , non gumming lube is what you want. Standard silicone is great. The best repair is to actually take the starter off removing the gear to clean the rust away before lubing, but you may get by doing it on the machine carefully as described above.

I had that problem (sort of) with a Craftsman snow blower/thrower.
It had the 110 volt starter on it. Every time i'd push the starter button, the starter would SCREECH LOUDLY! And not start.
Upon examination-i found the starter body wasn't setting squarely with the ring gear on the flywheel, but was hanging down at the back end of the starter motor area. This allowed the starter gears to not engage properly, which caused the high-pitched screech!
This seemed to be a factory goof. After i shimmed up the back end of the starter motor, the thing worked ok-no screeching.

I purchased an Ariens snow blower with a battery start instead of the 110-volt electric start, simply because it isn't unusual for a heavy Winter storm to cover us with snow and knock out our electricity. We live in a rural area.

Incidentally, I think you can rope-pull either model as an alternative to electric starting.

ZM