Worker control of production would only happen through government efforts though in todays society which is not libtertarian lol
Co-ops are "private socialism" and operate on the free market. Your local energy or grocery co-ops, credit unions, as well as notable large-scale retailer co-ops like Ace Hardware, True Value, and NAPA are all examples. The government didn't found them, nor did it establish them by forcibly converting a privately-owned business into a publicly-owned enterprise. A co-op can be founded like any other business, just with the power structure having democratic features and the workers (and often also customers) collectively owning the enterprise.
Incidentally, private corporations are themselves legal entities, and wouldn't exist without the government. They are effectively government constructs, though they are private sector entities and thus the government does not own or manage them (barring the occasional—and always partial and temporary—bailouts of failing companies, which are essentially a limited form of nationalization).
Edit: And libertarian socialism is not an oxymoron. Socialism, defined as public ownership of the means of production, can and does exist in free, democratic nations independent of the government, as the aforementioned example of cooperatives shows, and those espousing such forms of socialism can and often are opposed to "big government" and authoritarianism.