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No, a public good is not defined by whether anyone actually owns the thing. What makes something a public good is when you cannot exclude someone else from the use of that good.
Water and sand are not really examples of this -- especially not sand. I can go to a building supply place and buy sand. I can buy a beach too. I can buy a lake.
A public good is something where you can't prevent others from using it. A classic example is national defense. If the country protects my neighbor from an invasion from Canada, they have to protect me too. They can't just let me get invaded because I didn't pay my bill.
Public good is defined as a commodity whose benefits are available to entire community without restriction. Ownership of the commodity, or the facility that produces the commodity has no role to play in definition in of public good. Similarly the who caries out or pays for the activity resulting in creation of the a good, has no influence on whether it is a public or private good.
For example a program to eliminate the incidence of a particular disease, which may be carried out by a government agency is a public good, because such a program benefits everyone in the community equally.
A well kept garden which is part of a private house and is maintained by the owner is also a public good to the extent this garden enhances the beauty and attractiveness of the entire neighborhood. This added beauty can be enjoyed without restriction by anyone passing by or living in such a neighborhood.
Public goods are often the result of spillover benefits of activities of firms or people outside the marketplace.
Public good is the kind of good which lacks both excludability and rivalry.
Excludability means you can prevent others from using the good. For example, I can block other people from using my pencil because that is a private good under my possession.
Rivalry means your use of the good diminishes other people's use. For example, the fact that I reserve a front seat in the movie theatre means someone else cannot sit in the front row.
An example of public good would be street lamp. You cannot prevent a person from benefitting from the streetlight. You take advantage of the light does not mean other people cannot gain from that good, either.
These traits of public goods often cause some people to become free-riders. Free-riders are those people who benefit without paying any price for it. For example, you cannot prevent an individual from benefitting from national defense.
Remember the key points: non-excludability, non-rivalry, and free-rider problems.
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