At first I agree with @bbmaxwell. These "pawn giveaways" are more generally known as a 'gambit'. Gambits are known for sacrificing all sorts of pieces to get a better position. I don't usually accept gambits since they will lead to unnecessary tough positions giving you a rather small chance to win. Accepting a gambit might lead to an open king, a lost piece or any positional losses. Your opponents pieces might be more active, and your pieces will only be squished together. Like I mentioned before, it is completely unnecessary to take a single pawn/piece to lead to a bad position.
Why Do I Never See Good Players Go On A Pawn Rampage?
I'm really just beginning to learn chess, meaning more than just knowing the rules. I'm still terrible, and still mostly just going on intuition, so if I hit on something, I'll use that as a technique, until I figure out if it doesn't work.
One of my things these days is what you might call a pawn rampage. Not understanding why someone might let my pawn just sit there, able to take their piece, I'll take the piece, and then, if they assume (like most people) that I'm not enough of an idiot to just keep doing that, I just go on and keep doing that, taking two or three minor pieces down with it.
I'm sure the problem is that it hurts my position, though I'm not sure how much more important it is to have a pawn sitting in the center than it is to use that pawn to take out a couple pawns and maybe a bishop or knight. There's obviously got to be something more than just adding up the numbers involved, though I can't see what it is. But I also never see a good player do that. Of course, I also never play anyone who isn't rated around the same as me (and I've been hovering around 600.) So maybe a better player would never let me do that. But I just never see people do that.
You would have to show us examples. But if you've only faced players around your rating, then you aren't facing any good players either, (no offense). I it looks like your opponent is just giving you a piece for free, they probably are. Double check you aren't giving away anything more important yourself, but that's what usually determines games at your level. Both of you are giving away pieces for free, the winner is who gives away less while noticing and taking more.
#1 thing you should do at your level. Every move look and see what is not protected on the board, both yours and your opponent's, and see if it is currently attacked. You will find chances to take things for free. The next layer of this is to see, if something is attacked and defended, to count how many attackers and defenders, often there will be an imbalance and you can still take something for free or realize you need to protect/move a piece out of the way.
I could pull examples out of your games if you want to see what I mean by this.

Well you pretty much answered to yourself in your first post. Good players won't allow you to keep eating pieces with a pawn, because it's generally a bad play. Of course, just like with anything in chess there are exceptions to this rule, like sometimes people are willing to give up material (usually pawns) to speed up their own development. There are a lot of openings (called gambits) that are based on this idea, you give up a pawn (or even a minor piece) in order to develop your pieces faster and launch a quick attack on your opponent.
So, if you watch good players play, you won't see pawns going on "a rampage" eating multiple pawns or pieces, because it would be bad play. Either the person giving up pieces is giving too much material for no real compensation, or the person trying to gobble pieces with a pawn is actually just allowing the opponent to build a fast attack. Neither scenario won't happen, because anyone with a bit of experience will know that it won't work in their favor (and they expect their opponent to play well enough to not play bad moves just hoping their opponent responds with other bad moves).
In beginner games you see all sorts of strange stuff, because people don't quite understand the value of material, time (tempi of moves) and development properly. And they might just actually not realize their pieces are hanging. People also play a lot of objectively bad moves (one move threats) hoping that their opponent don't see their ideas, and sometimes they get away with it. This is called hope chess, and something one should avoid when learning the game, even if you sometimes get wins against other beginners playing like it.
Thank you. I guess I did answer my own question, or at least posed it as the most probable reason. I agree it's time to move beyond hope chess. I appreciate the responses.
Yeah. I'd also get in the habit of checking your games afterwards, or at least the ones you lost, and see if you can follow how you lost the game. If you don't see where things went south for you, or need a better idea, make a thread and ask about it.
Please don't be offended by this, but I think you have the cart before the horse. You never see good players do that because they don't get a chance to. 600 players typically have terrible board vision and leave pieces hanging all the time. Better players don't.
You mention that you're still going on intuition. Intuition implies a knowledge base of some kind; you're probably just guessing. Hit the Lessons, hit the Lessons hard and when you finish; do them again. My $.02. Best of luck to you!