Possible "reduced viral load" by those who are vaccinated simply doesn't mean that that load is below the threshold which makes somebody not infectious. In practice, it could, for example, mean that if such persons spend two hours in the room with you they can still transmit the virus to you, whereas the non vaccinated person with the virus would transmit to you in 15 minutes. So you can't just claim that a person is 100% safe because the load is just "reduced."
Additionally, for the immune system to respond, the infection has to happen first -- the virus has to spread through the cells of your body. We already know that the people are indeed infectious before their immune system response makes the symptoms. The delay in response has to exist even among the vaccinated people.
So what is sought after is a proof of sterilizing immunity, and there's no such still. I've read that the UK plans to evaluate the evidence for that in the following months by tracking the people who get the vaccine, which sounds good.
There is no threshold that makes people non-infectious. Or there is, and it is zero. Because even a single virus instance can potentially infect someone else. It is extremely unlikely to happen, but not impossible.
Because of this, for practical purposes, you either have to arbitrarily set a threshold at which someone is considered non-infectious, ignoring that it is not impossible for that person to infect someone. Or you have to stop talking in absolutes entirely and just talk about probabilities.
When taking about probabilities, one usually can recognize a reasonable threshold. In practice we do exactly that with most of the medicine: there could be some small chance that somebody can have health problems because his body responds unfavorably to the medicine, but if that chance is small enough it is considered acceptable when the potential benefits overweight the potential loss when the medicine is not used, its use is allowed. If the chance is big enough (i.e. potentially too many people will be affected) such a medicine is not allowed to be used, at least with the affected group of people.
The same is with the possibility that a vaccinated person infects somebody else. There is some point behind which it could be said that some vaccine has "sterilizing immunity" even if some small level of viruses could be present somewhere. For the current vaccine, the question is if the viral load in some point after the infection is decreased at all, and if it is, how much.
At the moment, however, it's simply not known if, in this case, Pfizer vaccine provides sterilizing level of immunity, if, then when, and in which percentage of the vaccinated. At the moment more or less we just know that the vaccinated are less probable to develop symptoms. Efficacy of 95% here means only one of 20 vaccinated develops symptoms when exposed to the virus, so we know that it's also probable that at least 1 in 20, even after being vaccinated, could be able to infect somebody else while being in the "pre-symptomatic" phase (as it is believed by the researchers that one transmits the virus before one's symptoms starts). We also believe that asymptomatic are also able to transmit. We don't know how much the vaccine affects the transmission that could occur when a vaccinated person is exposed to an infected one, and then later comes in close contact with other unvaccinated persons.
Additionally, for the immune system to respond, the infection has to happen first -- the virus has to spread through the cells of your body. We already know that the people are indeed infectious before their immune system response makes the symptoms. The delay in response has to exist even among the vaccinated people.
So what is sought after is a proof of sterilizing immunity, and there's no such still. I've read that the UK plans to evaluate the evidence for that in the following months by tracking the people who get the vaccine, which sounds good.