Once your immune system is trained to attack the known pathogen, it should be doing that before the virus is ever able to gain a foothold in your body and spread from you to someone else. People who have been vaccinated will not spread the virus.
Yeah, these claims are FUD. Yes, there are theoretical concerns; as with all new things, there is no complete certainty. But we should be absolutely clear that immunity usually reduces transmission. That that's at least very common is trivially observable in the real world too; no need for fancy statistics: just note that anything we have effective vaccines for has dramatically reduces spread in societies using those vaccines broadly.
Additionally, vaccine transmission reduction worries can be conflated with direct vaccine risk worries. There are risks to untested vaccine candidates, and some can even cause more severe illness. That's why we test, surely? And though I can't find the citation off the top of my head, there are even cases where a vaccine might enhance spread in extremely peculiar circumstances; I found a paper documenting such effect when vaccinating animals vs. an otherwise generally fatal disease; the vaccine simply delayed the inevitable and that delay caused a measurable increase in spread - but that's not at all like covid 19, which is not usually fatal. These theoretical concerns do not appear likely to be a factor this time, and most remaining doubt will clear up once phase 3 trials are complete: in particular, you'll know before you're vaccinated.
Whether the vaccines will reduce spread is obviously we don't know for absolutely sure; but reduced transmission does appear to be the most likely outcome. We should definitely act and plan as if it will (while measuring and hedging to detect if we're wrong). If new data appears that undermines this assumption, then we should adapt - but that's not likely. Much more likely is that the exact details of how much spread is reduced is unknown (so a 90% effective vaccine might reduce spread by less than 90%; or even by more; and we don't know those numbers).
So for example, when a paper like this https://www.thelancet.com/journals/lancet/article/PIIS0140-6... points out the limitations of currently knowledge, you should interpret that as just that - a limitation of current knowledge, not that it's very likely that a vaccine will leave spread mostly unaffected.
The final nail in the coffin for this FUD is that it's not actionable. Attempts to vaccinate - with the hope of reducing transmission - won't be ill-advised in retrospect in the unlikely event it turns out not to; it still reduces severity safely. So in terms of policy after phase 3 trials are complete and before mass-rollout: who cares? At worst vaccines will reduce disease impact with a good chance to reduce spread too. But if people start thinking the vaccine is likely useless, uptake could drop, which is throwing away such a great chance. This is classic FUD: doubt that's simply undermining rational action, without cause.
Once your immune system is trained to attack the known pathogen, it should be doing that before the virus is ever able to gain a foothold in your body and spread from you to someone else. People who have been vaccinated will not spread the virus.