Let's get the obvious out of the way first: your shield offers no protection from another Winston.
Winston has 400 health and 100 armor. The 100 armor will effectively halve the damage taken from the Tesla Cannon, but thankfully there are ways to burn through it more quickly than tickling with electricity: Winston's Jump Pack deals 50 damage when landing close to an enemy, which is almost half of the enemy Winston's entire armor, and his melee attack deals 30 damage.
Since most of your damage will come from your Tesla Cannon, landing at least one jump while the enemy still has their armor will give you a significant advantage, so learn the minimum distance you can leap and bait enemies into that range before jumping on them and firing your main weapon. If the enemy has little armor remaining, a swift punch before continuing to fire your Tesla Cannon is also effective.
The cooldown on the Jump Pack is almost exactly the time it takes to empty a Tesla Cannon reload if firing continuously: try to reload right before jumping, so you don't awkwardly stand around the enemy Winston while waiting for your weapon to reload and make yourself even easier to track.
Many Winston players will jump straight up and try to land on you, the best way to counter this is to jump half a second after they do and follow them on the way down, avoiding damage and stomping them instead. On the flip side, don't try to do this to the enemy, instead go for short hops that are hard to avoid.
Lastly, don't forget that if the enemy has 30 HP or less, a melee attack is faster at killing them than your Tesla Cannon. In a close match, this can be the difference between winning and losing.