Note: I think they prefer MMT to chartalist since the latter easily translates to charlatanist. Hee hee!
So here is what I think the MMT position is: Treasury debt is private savings in the sense that, to place the debt people need to be willing to purchase the bonds. Thus Treasury debt is money owed to private individuals. This is 2/3rds true - of the 13.5 trillion debt outstanding (Sept. 2010, according to the Treasury bulletin), 9.1 trillion is held by "the public." The remainder is in "Securities held by Government Accounts," or inter-agency debts.
Let's focus on the debt to the public. For arguments' sake, I'm go to abstract away from the level of foreign holdings (e.g. China) of U.S. debt and assume that all debt is held by U.S. citizens. This avoids having to discuss trade issues, which are not part of the story.
So in this set up, every dollar of public debt issued by the Treasury is ultimately held by U.S. citizens. You can own Treasury debt directly, or through some agent (e.g. mutual fund). Thus Treasury debt is de facto private savings. So far, so good - I have no quarrel to this point.
One question that comes up is: why are people willing to hold Treasury debt? Well, it does promise a nominal return (or real, in the case of TIPS), so the interest induces people to hold the debt. Also, Treasury debt is default-free, so it is less risky than other similar types of debt (including municipal, corporate and foreign debt). So, as a finance guy, it makes sense to put some Treasury debt into your portfolio. This default-free feature becomes very important when people are more uncertain than usual about the future performance of other bond issuers in general.
Now bonds are debt, and so must be paid back eventually. Every bond issuer has two options to pay back existing debt: use cash on hand (from retained earnings in the case of corporations, or from taxes in the case of governments) or refinance the debt (issue new bonds and use the proceeds to pay off the old ones). Refis are pretty common - corporations for example like to target a certain cost of capital and also look to minimize cost of capital, so they may shorten or lengthen maturities using refis when the markets are favorable. Refis are also how the Treasury operates; it typically just rolls over the debt by issuing new bonds to pay off the old ones (in addition to new ones to pay for more spending).
But if you don't have cash, and can't refi, you're in default. The Treasury is default-free because of the third option: printing money (I've discussed this before). So as long as bonds are denominated in nominal terms, the Treasury can never default.
Well, enough with the institutional details. When the Treasury issues a new bond, some people decide to buy the bond. So in a realistic senses, Treasury debt is private savings.
But what if no one wants to buy the bond?