Let me preface this by saying I don't own a MISTI and I have never used one, so I can't compare this to that stamping tool. I did however find the Memory MISTI to be outside of my price range so when the Precision Press came on the market I snatched it up. I do scrapbooking and card-making so having a tool that could be used for 12" x 12" papers as well as cards was key!
I understand some of the issues that users may be having. The paper can move or stick to the stamp if you have sticky ink and/or stamps. I have found that the clear stamps are more likely to stick to the paper and cause movement than rubber stamps. So, if I am using smaller clear stamps I will add a piece of washi to the paper to keep it in place or make sure I always push the paper up into the top right corner before I re-stamp.
I found that with large background rubber stamps, the end of the Precision Press that is close to you when you stamp is slightly higher than the end closer to the hinge. This means applying slightly more pressure at the closer end to make sure that the stamp makes even contact all over the paper. Also you can't place those large background stamps right up on the edge of the stamping area. For me these are minor issues and quirks of using the product.
All in all, a great product given the stamping options (clear, rubber, card fronts and 12" x 12" paper) and the price.