Escape from planetary atmospheres of the terrestrial planets in our solar system is dominated by ions in absolute numbers, as opposed to neutral particle species. Particles can be any type of molecule or atom here, mostly $\mathrm O^{+}$ and $\mathrm N^{+}$ for Earth.
For the case of Earth, a particle, once ionised in the upper thermosphere, can couple to the terrestrial magnetic field. From there, it will start orbiting in the magnetosphere, and it can be picked up by the solar wind at the magnetospheric bow-shock. Once transported by the solar wind, most of the species will be transported into interstellar space.
Your argument with the rings of material (not dust, see below) would be valid, if the velocity of the escaping particles would somehow be fine-tuned in a way as to stay in orbit. However, escape velocity from Earth is $14\ \mathrm{km/s}$, the orbital speed at $1\ \mathrm{AU}$ is $30\ \mathrm{km/s}$, so most particles will escape from Earth, not staying in orbit around the sun.
Add the solar wind on top of that, having typical speeds of $100{-}400\ \mathrm{km/s}$ and you get blown away into space pretty good.
There is of course a small chance that some of those escaped particles will be picked up on their way out by the other planets. This fraction is a function of the geometric cross-section of Hill-sphere of the encountered planets, but is still very small compared to the total escaping flux. I remember having read an article about a fraction of the atoms being lost by Venus, being later picked up by Earth, but can't remember the source.
This results in most gas ending up in interstellar space, as previously stated.
Assuming a constant average velocity of the solar wind of $\sim 100\ \mathrm{km/s}$, the solar wind can traverse the $\sim 150\ \mathrm{AU}$ until interstellar space in just about 6 years. From there, the lost particles contribute to the interstellar medium, although in negligible contributions.
Note that a 'dust cloud' is different from escaping atmospheric gas. 'Dust' in space are usually minerals. The most prominent ones are olivines, pyroxenes, forsterites, etc. that form macroscopic crystalline structures and could never escape by Jeans or hydrodynamic escape from an atmosphere, except via meteorite impacts.