Not implying, outright stating!

Colloquially private property means like "stuff I have exclusive or near exclusive rights to" but that's not what we mean when we talk economic systems. Something like your clothing, or that neat pot you made is personal property. Private property is a legal construct wherein someone is allowed to claim ownership over means of production, like "this field is mine, it doesn't matter if I'm using it or not, I have the legal right to control what happens there".

So an example of a market without private property might be something like:

• a field is held in Commons by a town
• a farming collective submits to work it
• they grow some potatoes
• the community recognises their right to the fruits of their labour
• they go to potato-lack town and sell some potatoes

Obvs that's simplified but it's a rough sketch of how farming used to work! you earned a right to use land by using it directly and what you grew on it was yours to do as you see fit (often a maniac with a horse and sword would take some portion first though. Because they're just better than you or whatever)