The rules say to do it without randomizing. This is from official rules (you can find the PDF file here, emphasis is mine):
Mission Team Vote: After appropriate discussion, the Leader calls for
a vote on the Team assignments. Each player, including the Leader,
secretly selects one Vote Token. When all players have their selected
Vote Token ready, the Leader asks for the Votes to be revealed. All
players flip over their Vote Tokens so everyone can see how they
Voted. The Mission Team is approved if the majority of Votes approved
and fails if a majority rejected the assignments; a tied vote is also
a rejection. If the Mission Team is approved, play continues in the
Mission phase. If the Mission Team is rejected, the Leader passes in a
clockwise manner and the Team building phase is repeated.
I actually don't agree with you that the second method makes much sense. You get way more information about the other players if you know how they voted in the past. Is player X always voting
No, is he voting
yes only if the player Y or himself get selected to the mission team? You wouldn't be able to tell any of that even after few rounds of play, and say, 10+ votes.
Another thing is convincig other players to vote for the team you want - if you know who voted
No, then you present good argument why you think they should change their mind, and they got nothing to reply you with, or say that they agree with you... Then everyone will see if they actually changed their minds or only said so, but voted
No anyway. If voting is secret, they could just listen to you, vote
No anyway, and they say "Hey, I voted
Yes, someone else must have changed his mind". And discussion can become dull.
I also think that randomizing votes would make things easier for the spies - they always should vote
no (they might just win the round with 5 teams voted down), unless the team selected is perfect for they needs. They would never have to think harder than that, and that seems less fun for me. That's why I would generally be against secret voting, but might give it a try a couple of times. Hope this helps.
If you're interested in the history of the game, I'm sure you can easily see that The Resistance originates from Mafia/Werewolves games. In those games, players openly suggest who to "linch" each day, and voting is (at least in most cases) open for similar reasons. In the prototype version of The Resistance, actually using your thumb down/up was used to vote
No, and this is the way I do it with in my group to this day.