I feel the need to post.
Amsah is mainly referring to Armada, the highest level Peach when discussing the matchup.
What most people in this thread are referring to when you discuss the matchup, is your opinion of Peach as a character, mixed in with your opinion of the peach vs sheik specific matchup. The only problem is, your opinions have been influenced, even if it is a subconscious influence, but nonetheless they have been influenced by what you've seen from Peach during your time in the Smash community.
And, what we've all seen is Peach being a good character, but losing to other characters who can abuse the game mechanics and her weaknesses vs her.
Even if you say your opinion of the matchup is not based off of what you've seen and been influenced by through your observations...And even if you say you are basing it off of your personal or discussed studies of Peach as a character, there's no way you haven't been influenced by the fact that there has been no pro Peach players besides Armada.
Most people (not everyone's) opinion of Peach vs Sheik is that Sheik wins, because of what we've seen so far from other pros playing, or perhaps, maybe what we haven't seen...We have never seen a Peach played at the level Armada's Peach is at. Just watch his videos and it's clear, Peach is much better than we've given her credit for.
We think we've figured the Character out a long time ago, and most people will just say, "Armada is just really good, Peach isn't better as a character."
The fact is, Peach is better than we thought she was. No one knew Peach could edgeguard that good. It's not just Armada, anyone can do that stuff if they learn what he's learned. Keep in mind, trying to copy him will not suffice. He most likely knows every single option Fox has to recover at any time, and also bases his edgeguarding not only off of that knowledge, and reactions, but also based off of what he thinks his opponent is going to do.
If someone put in the work Armada did for Peach and were creative enough to think of the things that he does with the Character, Peaches would be doing just as good as Foxes. That is a personal belief that I will stand strongly by. I mean, let's take a look at some of Armada's matches.
First off, if you are able to analyze matches even at least decently, you can see that Peach, played at the highest level of play we've seen so far, is just a walking (or I should say, floating) hit box of death, that can cover nearly every option and make herself nearly impossible to hit. She can adjust her spacing so well with floating, and combined with her nair, her Fair and lagless float cancels, she can protect herself from any incoming approaches.
If you do manage to hit her, you're going to tie a lot of the time since her attacks have high priority, and depending on the percent, you can either get thrown into an edgeguard position, or just have to tech and reset.
Her attacks themselves aren't really deadly like Marth's Fsmash or Fox's Usmash, but the fact that they can connect at nearly any percent to tons of other moves makes them just as good. The fact that Peach can chase you anywhere with floats on platforms, or if she can't get there in time, she'll be close to you in a zoning position where one mistake will knock you off the stage is just too good.
I'm just talking about her general moveset. Specifically, let's talk about other amazing aspects she has, like her turnips. I'm not going to pretend like I know what Armada is doing specifically with his turnips, but it's pretty obvious that he knows when to throw them, and not only what options their opponent has to deal with them, but how to counter those options as well.
Her out of shield game is stupid. If you mess up shield pressure by misspacing by 1 inch or a using the move a fraction of a second too late or early, you're getting naired out of shield. If you use a laggy attack, you're getting fc bair'd out of shield or nair'd. Even if the range is far.
What about her Dsmash. Yeah, it's not that hard to beat, but the fact that if you mess up once against it you could take up to a bajillion damage, or how it can stop tons of attacks on the ground.
I could go on, but in short I think what I'm trying to say is that Peach can definitely beat Sheik at highest levels of play. I don't have a specific ratio in mind, but in Pal I def think she has the advantage, and in NTSC I see it going either way.