So, do you either use nominative + mellett OR adessive? One or the other? Which do native speakers use more often?
It is mostly a taste of the speaker. "telefon mellett" and "telefonnál" is just the same in most cases. I myself use the two nearly the same.
Some extra insight: if you use a person's name to describe a place, you always use "-nál/nél", because otherwise you would say you are beside the person. Like "Péternél várlak" means I am waiting for you at Péter's (place), while "Péter mellett várlak" means I am waiting for you near Péter (the person). :)
I think the more direct and actual a location is, the more freedom you have to choose between "mellett" and "-nál/nél". As locations, designations shift towards the abstract, the more and more the "-nál/nél" option comes forth.
But in everyday language, it is just the same and varies on the speaker. Yes, it has slight differences in meaning, but that is mainly an agreement between the parties of the conversation, like what radius, vicinity or side does "mellett" and "-nál/nél" mean.
I think the difference is between a fixed location and a relative location. And some functional purpose involved.
"-nál"/"-nél" - I am specifying a location. You know where the telephone is; I will be there. You know where Péter lives; I will be at his place. Etc.
"mellett" - I am specifying my relative position to some object or person. Wherever the telephone is, I will be next to it (I am on call). Wherever Péter goes, I will be beside him. Etc.
And when the object is usually stationary, like a house or a bookshelf, then the two meanings blend together. You can use either one.
I see you know German, too. Would it be correct to say that "nál" can be translated into "bei", whereas "mellett" can be translated into "neben"? I'm Dutch and we use these words too ("bij" and "naast")