Rogue has a better winrate going second for sure, but all other classes have a slight advantage going first. Don't let the occasional coin power-play convince you otherwise. ESPECIALLY in arena, whoever is going first is going to put down the first minion most of the time, which means they're going to have initiative until the second player does at least one more 2-for-1 card trade than the first player.
It's not really that simple.
Let's say it's Priest vs Paladin. Priest turn 1, doesn't put down anything because most of the time players do not have a good 1 drop at turn 1. Player 2 doesn't put down anything. Player 1 puts down a 2 drop, player 2 puts down a 2 drop. Player 1 puts down Dark Cultist (the best 3 drop essentially) and trades his 2 drop into the other player's 2 drops. Player 2 coins out Truesilver Champion, takes out Dark Cultist and player 1 is now at a distadvantage because whatever they throw down is most likely going to be killed (anything short of a Yeti basically).
This happens way too frequently. If it's not that it's usually Mage turn 1 coin Snowchugger, player 1 throws down a minion and then the Mage uses Frostbolt and keeps the pressure up.
Another common 1st turn coin drop play is drop some random 3/2 drop as Paladin, then next turn Argent Protector the 2 drop to keep him alive and maintain board control. This snowballs over the coarse of the game unless the other player has AOE or some big tempo swing card.
All of this is amplified greatly by so many sticky minions these days where it's very important to have the first big play on board. That's why a lot of Arena matches are decided early by a big power play usually with a coin like a turn 1 minion into coin Velen's Chosen.
Turn 1 has huge advantage however if they have something like a Zombie Chow or a stealth worgen vs the other person's 3/2. Especially true if it's like Zombie Chow into Argent Protector, absolutely devastating. But there aren't as many 1 drops in the game to make a difference off the bat... it's usually a 2 drop combo or a powerful 4 drop with the occasional tempo swing cards after turn 4 like a turn 5 Fire Elemental.
Usually if the turn 2 player doesn't have a good use of the coin then they are at a disadvantage of course because turn 1 player will put down the bigger minion first. But the coin does a lot in smoothing out one's curve. You can get by not having a 3 drop going 2nd because of the coin but as a 1st player if you don't have a play there then you are at a disadvantage (which used to happen a lot but not as much now).
