You can't really compete with the folks who are doing bulk REOs, and you can't do anything about investors who aren't buying, but if you have buyers who are going around you -- directly to the MLS -- you need to figure out a way to get better deals on the same properties they're trying to get to directly.
For example, I just underbid another investor on a property by $5K, but because I offered over $20K in earnest money and no contingencies, the bank accepted my offer. If I wanted, I could wholesale the property to the other buyer for a $5K profit, and everyone would likely be happy.
Another important aspect of getting your offers accepted (over other buyers, and then wholesaling to them) is to make friends with the REO listing agents. Get to know them, take them to lunch, etc. If you're good to do business with AND they like you, they can often push through deals that otherwise wouldn't get accepted.
For example, I had a listing agent the other day who told me that the bank had countered my offer. She basically then said, "If you choose to stand firm on your original offer, I think I can push it through for you." I stood firm, and the bank relented and accepted my offer. If I weren't on good terms with this agent, I likely wouldn't have gotten the deal anywhere near my original offer price.