Rarely to 0%, but the reductions are still dramatic.
Most creditors agree to "concession rates" of 6% to 9% through a DMP. Some creditors occasionally offer 0% or near-0% through DMP programs, but it's the exception rather than the rule. The standard outcome is a rate that's roughly 60% to 75% lower than what you're currently paying, not a full elimination of interest.
The specific rate you get depends on the creditor, not the counseling agency. Each credit card company has pre-negotiated DMP concession rates. Discover might offer 6.99%, Chase might offer 8%, Capital One might offer 9.9%. Your counseling agency doesn't negotiate these rates on a case-by-case basis (they're established agreements). So the agency you choose matters less than which creditors you owe.
Even at 8% instead of 0%, the savings are enormous. On a $20,000 balance, dropping from 24% to 8% reduces your annual interest from $4,800 to $1,600. That's $3,200 per year in savings, or about $267 per month that now goes toward paying down principal instead of enriching the credit card company.
If you truly need 0%, a balance transfer card offers exactly that for 15 to 21 months. But balance transfers require good credit (700+) and have transfer fees (3% to 5%). A DMP has no credit score requirement and much lower fees. For most people with significant credit card debt, the DMP's 6% to 9% rate across all cards beats the balance transfer's 0% on one card.
Don't hold out for 0% if a DMP can get you to 7% or 8%. The difference between 8% and 0% on $20,000 is about $133 per month. The difference between 24% and 8% is $267 per month. The DMP already captures most of the benefit.