Alcohol does not necessarily affect all people who stutter in the same way. A search through the PubMed database reveals that this question has not been researched extensively, but some clients of the clinic have anecdotally reported that their speech becomes more fluent after having a few drinks. 

Self-medicating for a stutter with alcohol, marijuana, or any other drug is not recommended. Although people who stutter are no more likely than the population at large to have substance abuse disorders, people who experience anxiety about their stutter may be more at risk of substance abuse disorders. This is because people with anxiety disorders are at an elevated risk for substance use disorders. https://pubmed.ncbi.nlm.nih.gov/20643799/