Are we talking about ...oh, what should we call them... how about "Occasions?" Anyway, are we talking about Occasions which are randomly selected at the beginning of the game with the Kingdom cards and the Events or are these things that would get shuffled together and one randomly flipped at the end of each round?
I feel like one random effect that lasts the game could be interesting. All it changes is how you play that game. Say everybody starts each turn by drawing a card. You know you have one more card every turn, so you will probably get to Provinces easier, but you also know that everybody else has the same opportunity. So you know how it could potentially affect you and you build your starting strategy around that, just like building your starting strategy around whatever random 10 Kingdom cards were dealt in a standard game and any Events that show up.
On the other hand, randomly revealing one effect after each round doesn't match evenly with the current game mechanics for my liking. Other than being able to look through the pile at the start of the game, you won't know what's going to happen each round, and that makes the luck factor more important. It might be more chance-based than just shuffle luck, and that's already a lot of luck. It also probably massively increases the first-player advantage; the first player knows what the Occasion will be on the other players' turns and can plan attacks accordingly (do I play Masquerade or Noble Brigand?) while the other players won't know what the Occasion will be on the first player's turn. Making Occasions like this would be very difficult, because you would have to devise Occasions that won't mess with the player-order advantage, are not political, and have a balance between harmful and helpful for most strategies - in the current space, that will be very difficult.
It's a nice idea, something like events that you don't have to use a buy on, but I'm not sure how I feel about it unless I get a more descriptive example.