It depends on the recipe of course, but most of the time the leavening effect is slight. I often leave it out- for instance, most of the time in chocolate chip cookies- because I like them with a little more tooth. More cakelike types of cookies, it may be fairly essential.
It acts as a leavening agent, but also helps prevent spread and does even more! Here's a great article breaking down the different things it does for your dough: http://www.seriouseats.com/2015/12/cookie-science-how-baking-soda-works.html
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My fave cookie this season was an easy, versatile 'Cook's Illustrated' icebox cookie with no leavenings, no spread, and a flat, crisp finish.