In Gloomhaven, can you get the effects of winning a scenario twice?
It depends on which effects you're referring to. You always get the experience (including the base XP depending on the scenario level), any loot you've picked up, as well as your checkmark or two if you've completed the battle goal, and any progress you've made towards your personal quest.

You do not however get any of the scenario rewards that are listed in the scenario book. This is because after beating the scenario, it can only be played in casual mode, which doesn't grant these rewards. You also can't loot any numbered treasure tiles if you've done so before (however, if you missed the treasure the first time around, you can replay the scenario to grab it).

Relevant sections from the rule book:


  Numbered treasure tiles can only be looted once. After they have been looted, they should be crossed off in the Scenario Book as a reminder. — p. 15





  In addition, once a scenario has been completed in campaign mode, it cannot be undertaken again in campaign mode by any party. — p. 34





  [In casual mode,] Players can still gain experience and money, loot treasure tiles, complete battle goals, and make progress toward completing their personal quests, but any story text or rewards listed at the end of the scenario are disregarded. A party in campaign mode can switch to casual mode to go through a scenario they have already completed [...]. — p. 34