Okay this post went on a little longer than I intended, partly because social ownership is a fairly broad concept, but I hope it answers some of your questions:
The point of social ownership ought to be that the people who either work in or directly rely on the organisation are the people who have meaningful control over it. This immediately excludes any kinds of pure owner who may never interact with the organisation other than for having decision making power over it and extracting profits from it.
Where worker co-operatives are concerned, an important aspect is that workers should have equal say (in the sense that voters in a democracy should have an equal say) over their organisations. Where corporations have 'one share, one vote', worker co-ops are 'one worker, one vote'. Of course in practice the nature of this varies - in smaller co-ops, the workers tend to have a more direct say over the operation of the organisation and in larger ones are more likely to just be able to elect board or committee members at the overall level, and more direct decision making for their organisational unit. All of this happens while workers retain all the same rights and privileges they'd be afforded at a traditional corporation such as the ability to leave and change jobs if they want to, so they get to have their cake and eat it in this respect but the cost is arguably more responsibility in the form of taking informed decisions.
A couple of examples of longer lasting worker co-ops:
https://en.wikipedia.org/wiki/Mondragon_Corporation - large international co-op from the Basque region of Spain, in business since 1956
https://www.suma.coop/ - an equal pay worker co-op which has been operating since 1977
There are also other kinds of co-ops such as consumer co-ops where anybody who shops at a co-op can become a member. You also get community co-ops where residents directly, jointly own their own local accommodation or amenities. Some co-ops are hybrids.
The co-op group in the UK is a large, long running group of consumer co-ops which have been going since the 1840s: https://en.wikipedia.org/wiki/The_Co-operative_Group and they have nation-wide businesses including retail, energy, funeral services and so on.
Sometimes it may be one of the above types of ownership or a combination. Co-ops usually adhere to a set of principles called the Rochdale principles: https://en.wikipedia.org/wiki/Rochdale_Principles which define various rights for members and ultimately a culture that co-ops should adopt.
Building societies and credit unions are another form of social ownership where those saving with them are the members:
https://en.wikipedia.org/wiki/Nationwide_Building_Society - nationwide is a UK-wide building society dating back since 1846
Government control, where there is a functioning democracy, can be a form of social ownership. It is just a very dilute kind. It's easy for your voice to get lost in a sea of millions, but that does not mean it isn't there - things like FPTP and the electoral college notwithstanding... Everything owned by a democratic state would be "sort of" social ownership; just a very weak, undesirable kind that puts too much power in the hands of a small group of representatives. Nevertheless, there are certainly some cases where it makes sense such as things which are natural monopolies like water sources, rail networks and indeed in the UK the NHS enjoys wide public support.
In that respect, municipal/local council control is also a form of social ownership but is much closer to home. Municipal libraries, public parks, public restrooms and such for instance are very much in keeping with the principle of social ownership. Municipal socialism as a specific aim has become a goal of some places, for instance in late 2018 Preston was named 'most improved city' in the UK after following a programme with this aim ultimately by leveraging local institutions to improve the area: https://www.theguardian.com/politics/2018/nov/01/preston-nam...
When it comes to social ownership I think sometimes something gets lost in translation, like you get some people who seem to think that just nationalising everything fits the bill - but the truth is that the model you have to choose has to fit the problem the organisation is supposed to be solving at the right scale. Having your local park managed by a centralised state authority by somebody who has never visited is probably a horrible idea in pretty much the same way that having it managed by somebody who only wants to profit from it but has also isn't particularly attached to its use as a park for residents may be a horrible idea (or certainly would be seen as such by those residents, at any rate). All of these approaches are appropriate tools for the goal of enfranchising communities and workers to have decision making power over things that affect their lives provided they're used right.
Nevertheless, none of this matters if you're otherwise operating within the confines of some highly authoritarian nation-state that in practice can totally take the reigns of your organisation, effectively disenfranchising members or constituents from making decisions.