I understand what a transformer is, but even without architectural changes they could probably utilize source-trustworthiness (if it was provided during training) to give better responses. During training they would learn that trusted sources help better predict factual answers where that is called for (i.e. in context of a WikiPedia page).
Personal experience would require architectural changes, not least online learning. No need to anthropomorphize - our brain is also just a machine, and we can use it as inspiration for what is missing to build more intelligent systems.