Seeing that certain symbols can have multiple meanings depending on the user (a good example is the Swastika, used in various cultures across the world), and the cross (which can represent the Christian Faith as well as certain secular countries, and possibly other meanings too when shaped as a sword), who determines what a particular symbol means?
It would seem to me that the only fair answer would be its user. A good examples is literature. Who determines the true meaning of the text? Of course it's the author. The rest of us can only speculate.
It depends on the situation but generally speaking, 50% of the arts meaning is based on the artist and 50% of its meaning is based on the viewer. To make sure the symbol/logo get close to a 100% connection to it's desired meaning, the artist must know how to connect to the targeted audience and predict how the audience will view that logo/symbol.
The artist generally designs something to send a certain message to the viewer, however if the majority of viewers don't get that message and instead, see something different.... then in a way, the artist has failed in sending the message he or she intended and that symbol or art then takes on a different meaning.
But as I said, it depends on the situation, as for example, a musician makes a song (which can end up being a symbol of a movement, era, identity, etc.) and wrote words a certain way to mean something in particular.... yet not everybody will get the same meaning from that song or its words..... and yet to add to that, some artists will write a song meaning one thing, but then not actually explain the real message, thus leaving it up to the listener to determine for themselves what the message is, which can be endless in different meanings.
In regards to a logo/symbol for a company or organization, it is usually the company/organization that determines which symbol/logo to go by that will identify themselves at a quick glance. Some are designed better then others..... but overall, whatever symbol they choose to represent them, they must ensure they know what attracts their targeted audience.... thus be aware of cultural interests and dislikes.
As an example, let's say I own a auto repair shop and I choose my corporate identity/logo to be a fluffy teddy bear holding an ice cream... does that logo catch your eye?
Perhaps.... you might even easily remember it..... but does that logo actually tell you what my company does and does it even relate to auto repair?
Not at all.
The design one uses to identify themselves in a graphical sense is generally designed and created by the company/person using it and in the end, they determine what the final product will be, how it will be used and where to use it.... but it's true meaning, unless explained by the owner/user of that logo/symbol, is left up to the viewer to determine for themselves if it sends the right message its supposed to, or if the logo/symbol actually works.
However if a logo/symbol doesn't send the right message from the start, if a company or organization continues to use that logo/symbol for a period of time, eventually they can condition the viewer into knowing that logo/symbol means that company/organization and what they do.
Take the MacDonald's logo... a big yellow M on a red box.
Does it actually tell you what that company does or the services it provides?
Not at all.... but everybody has seen it for so long.... pretty much all through our lives we've seen this logo in one form or another and we have been conditioned to know that logo means MacDonald's, and their service is fast food.
There's even more one can explain to your question, but to say the least, the question you asked does not have an easy answer..... but rather, multiple answers depending on the type of symbol/logo you're asking about.