I am far from an expert on braking systems, so take the following for what it's worth....
To my knowledge, there really wasn't an "automatic" vacuum system to correspond with the more common straight or automatic air brakes. The Eames vacuum brake (the most common type) operated comparably to a "straight air" system. It was a simple and cheap technology, which is why so many narrow gauges used it. Briefly, steam under pressure was ejected through a venturi connected to the train's brake line creating a vacuum in the line which acted on a diaphragm on each car (usually the engine as well). The diaphragm was connected to the brake linkage, and acted by pulling on the linkage rather than pushing as the Westinghouse system did.
The chief advantage to the Eames brake, aside from low cost & simplicity, was that brakes were instantly available any time steam was up. There was no lag time between brake applications as with air brakes where the air reservoirs must be refilled after release before brakes can be re-applied. The disadvantage was that there was no fail-safe feature. A reduction in steam pressure, vacuum leakage in the line, friction on the brake rigging could all have a deletrious effect on stopping ability. The system was not considered adequate for trains of more than ten cars.
Some later versions of the Eames brake placed the ejector nozzle in the smokebox, which might explain the apparent lack of a muffler.
If you happen to have a copy of Hilton's American Narrow Gauge Railroads there is a fairly good description of the Eames system, with machine drawings, beginning on Page 187.
J