Yes, according to Laws and Customs, the Noldorin Father-name was public.
And according to the Shibboleth, Gil-galad was an epesse.
I've constructed the following primarily based on
two accounts from
The History of Middle-earth. It's not entirely clear whether the
"Note on Mother-names" [the later if briefer text in any case] is intended to replace "Of Naming" though I note in advance that the Chosen-name could be a Noldorin custom which differs from other Eldar.
Also, even according to Christopher Tolkien the concept of the Chosen-name [this "probably Noldorin" type of chosen name] might have been abandonded [if not merely left out due to brevity].
If any dare cross the line of sleep.
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzz
Of Naming from Laws And Customs, Morgoth's Ring
The description begins with respect to the naming of Children among the Noldor.
Father-name It was the right of the father to devise this first name. It remained unaltered save for such changes as might befall its spoken form in the passing of the long years (even the tongues of the Eldar were subject to change).
Chosen-name In which point, maybe, the Noldor differed from the other Eldar. It is said here that the Elf-child had the right to name himself or herslf. The ceremony of Name-choosing could not take place before the child was deemed ready and capable of lámatyáve, as the Noldor called it: that is, of individal pleasure in the sounds and forms of words. In elder times the Chosen-name or second name was usually freshly devised. In later ages, when there was a great abundance of names already in existence, '...it was more often selected from names that were known. But even so some modification of the old name might be made.' JRRT
This wording does not seem to imply (in my opinion) that it was absolutely necessary that some modification must be made to the Chosen-name, but that it "might" be made. In any case the 'true-names' are here said to be the Father-name and Chosen-name, and the Chosen-names were said to be regarded by the Noldor as part of their personal property. New chosen names could be added.
Anessi given or added names.
Here we have the
Mother-name, a name of insight or of foresight. A Mother-name was given in the hour of birth or on some other occasion of moment, indicating some dominant feature of the Elf-child's nature as perceived by her, or some foresight of its special fate. When solemnly given a Mother-name could also be regarded as a true name and was sometimes placed immediately after the Father-name. Mother-names of insight in general use sometimes replaced the Father-name and Chosen-name, however the Father-name and the Chosen-name (among those that had the custom of essecilme that is) remained ever the true or primary name, and a necessary part of any full title.
Other Given-names were not considered true names, and names or nicknames of this kind might be given by anyone... in memory of some deed, or event, or in token of some marked feature of body or mind.
Note On Mother-names The Peoples Of Middle-Earth
The Eldar in Valinor had as a rule two names, a
Father-name and a
Mother-name. Mother-names were given later, often some years later, but also sometimes soon after birth.
Epesse 'After-name' a nickname not necessarily given by kin and mostly given as a title of admiration or honour. Later some among the Exiles gave themselves names, as disguises or in reference to their own deeds and personal history: such names were called
cilmessi 'self-names'. The true names remained the Father and Mother-names it seems, according to this account.
This is the brief version anyway
