While there is a Product:Toy:Pogo Stick category, when talking about fuel-powered pogo sticks and transportation, the "toy" category is not really appropriate. A couple items in that category (linked) should probably be moved to this one.
The fundamental compression/explosion aspect of an internal
combustion engine is ideally suited for pogo sticks, since there is no need to convert reciprocating motion into rotary motion. However, the 2-stroke combustion cycle used by most fueled pogo sticks is environmentally dodgy (skip next paragraph if you already know why).
In the conventional 2-stroke cycle, a fuel-air mixture is compressed. Then a spark ignites the mixture. At the end of the expansion/power stroke, openings in the side of the engine cylinder allow hot gas to escape, while simultaneously allowing fuel-air mix to enter. Pollution can occur because some of the fresh fuel can also exit the exhaust port.
A Diesel version of the 2-stroke cycle (one of the linked Ideas) could theoretically be cleaner, since fuel is added separately from the fresh air. But in practice the art of fuel-injection is imperfect, and not all the fuel burns cleanly, leading to the typical black smoke emitted by Diesel engines.
The 4-stroke combustion cycle can definitely be cleaner, so that is what I'll describe for this Idea.
The above ASCII sketch (ignore the underscores) tries to show (overhead view) how the central column of a pogo stick should be associated with 4 engine cylinders. From underneath, the central column would have 4 piston rods attached to it; all the pistons would go up and down together in the cylinders. I'm specifying 4 cylinders solely for reasons of Dynamic Balance; 2 could work if you didn't care about such things.
I'll talk about 2 in this paragraph just to show how it works. One cylinder has fuel-air mix getting compressed, while the other cylinder is doing an "exhaust stroke" --a valve is open allowing the waste gas from the previous cycle to be pushed out. Then the exhaust valve closes and an intake valve opens for that cylinder, while the other cylinder gets the spark to ignite the compressed fuel-air mix. The power stroke in that cylinder allows the other cylinder's piston to suck fresh fuel-air mix. The intake valve closes, and now this description repeats, with one cylinder doing compression and the other doing exhaust.
In the 4-cylinder engine, 2 would be compressing and 2 would be exhausting, and so on. In the ASCII sketch, Dynamic Balance is achieved by pairing the cylinders on directly-opposite sides of the central column (such as lower-left with upper-right). The cylinders in each pair are always doing the same part of the combustion cycle together.
Next, the art of electric-powered engine valves has been developed to the point (linked) where that would be the most practical way to control the valves for this engine.
The engine can be air-cooled, since we all know a pogo stick won't be doing very many combustion cycles per minute (remember that air-cooling can work for, say, lawnmowers, which may have 1000 combustions per minute). The cylinders can also be smaller-diameter than the single cylinder of already-existing fuel-powered pogo sticks, simply because we get power from 2 at a time, and we don't really want the pistons to come all the way out of the bottoms of the cylinders, with the rider and the tops of the cylinders going unsafely high.
(The math says that if 2 smaller cylinders have the same interior total volume as 1 larger cylinder, and their lengths are the same, then the diameters of those 2 cylinders can be about 30% less than the diameter of the 1.)
Finally, we add something to the top of the pogo stick, known as a "tuned mass damper" (linked). Under electronic control and moved with electric power, this can ensure that the overall pogo stick remains generally upright (like a Segway), and doesn't tip over too far, as the rider bounces his or her way down the road.