Based on the wording, and the "no visiting" rule, I'd say that using Beggar's reaction triggers two events - one being "Gain a Silver" and the other being "Gain a Silver, putting it on top of your deck". But since those events trigger simultaneously, you get to choose the order in which they happen, and as each one resolves you can react to it. So you can either choose to resolve the normal gain, then resolve the on-deck gain and as it happens react to that with Trader - or you can choose to resolve the on-deck gain first, react to it with Trader, then go back and resolve the normal gain.
It would be similar to Stonemason, which specifically rules that the gains happen one at a time in an order of your choosing, which you could use to do something like Stonemason on a $5, gaining Rats and another $4, choose to gain the non-Rats card first, use Watchtower to top-deck it, then gain the Rats and use Watchtower to trash it, drawing the card you just top-decked.