People absolutely could and did brew their own alcohol and sell it on a smaller scale. In fact, I'd argue that most 'bootlegging' which took place was of this variety, rather than the more glamorous Al Capone organized crime variety.

Although authorities routinely uncovered massive stills which could produce thousands of gallons of alcohol, they also frequently encountered smaller stills which operated in people's homes. These stills would be run either for personal consumption, or as businesses. In the countryside, farmers would produce their own liquor and then give or sell the excess to their neighbours.

In cities and towns, since the saloons were closed down, many enterprising individuals filled the gap in the market by running drinking establishments in their own homes, serving liquor they brewed. These were incredibly common. In Prohibition-era Pittsburgh, authorities estimated that most liquor was produced in these small-scale operations, not in the huge enterprises. Running a drinking establishment from a home was also a way for women to make some extra money during Prohibition, and many such establishments were run by women. In other cases, saloon-keepers who had moved to selling ice cream and sodas after Prohibition would continue to illegally sell alcohol, and they would likely operate as 'wholesale' purchasers of illegally brewed alcohol who were not necessarily involved with organized crime. Illegal home brewing was particularly common in immigrant neighbourhoods, most of which would be formed from European immigrants who had brewed alcohol back in their home countries.

This is a pretty short answer, but I just wanted to convey that most bootlegging and violation of Prohibition was committed on a very small scale, by people simply looking to get by and make an extra buck.

Two interesting articles which talk about this sort of thing are:

Mary Murphy, "Bootlegging Mothers and Drinking Daughters: Gender and Prohibition in Butte, Montana" American Quarterly 46:2 (1994).

Julien Comte, "'Let the Federal Men Raid': Bootlegging and Prohibition Enforcement in Pittsburgh" Pennsylvania History 77:2 (2010).

See also:

Michael A Lerner, Dry Manhattan (Cambridge: Harvard University Press, 2007).