Socialism is when the *workers* own the means of production. It's something akin to a worker co-op. It's similar to owning your labor under capitalism as you put it because you're still free to leave the co-op if you can't reach an agreement on terms of employment, and you're still entitled to your share of the profits, probably on top of some base wage. This means there's no oligarchical capitalist class above the workers taking an outsized portion of the profits, which socialists and communists view as parasitic rent-seeking behavior. Notably this explicitly ties your gross pay to the success of your specific co-op, which means while you may make the same amount as other workers at the same co-op, but there is NO way that EVERYONE makes the same amount because different co-ops will face varying levels of success. Communism is when the *community* or *society at large* own the means of production. In practice, this effectively means that the state owns them. In theory, it should mean in accordance with the will of the community/people, but actual communist governments have historically leaned more authoritarian, in which case, it's closer to as you say where the state owns your labor. However, personal property is not necessarily the means of production, like a TV is primarily for consumption rather than production for most people, so while it may be difficult or impossible to own certain things, nothing intrinsic to communist theory prohibits personal property entirely, though some may still advocate for it.