Having the vaccinations may well stop you catching the virus (I sincerely hope so!) or getting ill with it but I don't see it entirely stops you spreading it. Think back to the start of the pandemic when we were all wiping our groceries and anything we touched. Does anyone do that now? If you touch an infected surface you will still be spreading it around, vaccine or not!
There seems to be a sea change in the official thinking, a move towards 'everyone has been vaccinated so now we just have to live with covid much like we live with flu.'
So of course it will still spread - it's a air-borne virus so how can it not?
It would be nice to get rid of masks and see people smiling again though. (Not sure that will happen anytime soon.)