Democrats want to expand Medicaid and to do so with a blank check, an open ended federal expense. Under the ACA federal funds will ultimately pay for 90% of the expansion under Obamacare. The states choosing to expand eligibility pay the other 10% plus, of course, their normal share of Medicaid before Obamacare.
If costs become a burden, reducing the fees paid to providers or some forms of rationing of care (the Oregon plan) will likely be employed possibly further lowering the quality of care received by beneficiaries.
Republicans want a defined contribution approach (used by some large employers for their retirees) by using block grants to the states where the increase in those grants may not equal the actual health care inflation. The goal is to limit the financial exposure to the federal budget.
As you can see, saving money is all about cost shifting in some way.
In both cases we are ultimately likely to get to the same point unless we accept that the growing cost of Medicaid should not be limited in any manner. The problem for the states is the drain on their budgets and/or a growing burden on state taxpayers and the possibility of reducing other state services.
If you want more details here are some good links.