I have done both. A significant strike through 1976-P Lincoln resides in the complete set. A TX quarter with a wire strike through is in a PCGS MS67 slab.
For me, it depends on the rarity of the error. If it's minor, stash it. If its a major error like struck on wrong metal, major doubling, double struck, rotated die, a mule ? .... I'd get it graded and authenticated. But, in reality, it's really a personal choice and since errors cost forty bucks for grading, a minor error wouldn't be worth the fees for grading, in my opinion.
I keep all my error coins separate in 2x2's and in those silly red boxes they sell at the coin shows. So if I had that one I'd break it out and put it in my error collection. If you don't do that eventually you'll forget where and what it is. I would but then I'm just old and forgetful. Always rememeber the 3 signs of old age. Forgetfulness and I can't remember the other 2.
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But, in reality, it's really a personal choice and since errors cost forty bucks for grading, a minor error wouldn't be worth the fees for grading, in my opinion.
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San Diego, CA