There are 3 factors that determine how much a vaccination reduces spread.
Vaccination reduces spread significantly because the vaccine is highly effective at preventing infection, and you can't spread a virus that you don't have.
Vaccination also reduces spread in breakthrough infections because a shorter duration of infection reduces the opportunities for the virus to spread.
Studies show mixed results.
"the duration of illness was shorter, with 2.3 fewer days spent sick in bed" https://www.nejm.org/doi/full/10.1056/NEJMoa2107058
"Vaccination reduces the risk of delta variant infection and accelerates viral clearance."
NOTE: For data to backup claims of effectiveness vs infection, see Effectiveness vs Infection