Now to attempt to explain the semantics of what could be a confusion regarding the use of "publicly" in reference to hermit profession of the evangelical counsels. Canon Law 603 effects the word "publicly" in regard to consecrated Catholic hermits who profess the evangelical counsels, as cited: "publicly professes in the hands of the diocesan bishop."
For consecrated Catholic hermits who do not discern nor request of their bishop the CL603 option, their profession of the three evangelical counsels can be made public or kept private in the normative, contemporary, semantic meaning of "public" and "private".
While vowing to live a life of poverty, celibacy and obedience is a required inclusion in all consecrated Catholic hermits' vows, the public or private nature of these three avowed counsels are not one and the same as hermit who, per CL603, "publicly professes in the hands of the diocesan bishop."
[CL603: "...publicly professes in the hands of the diocesan bishop the three evangelical counsels, confirmed by vow or other sacred bond and observes a proper program of living under his direction."]
It is this stipulation in Canon law (CL603) that makes such profession to be termed "public, " as in being recognized by law. The hermit's diocesan bishop is also, as a stipulation of CL603, to direct the hermit's "proper program of living".
Catholic hermits whose professed vows [recently and ostensibly coined "private profession"] include the three Evangelical Counsels, may profess these to a priest, abbot, abbess, spiritual director, spiritual friend, family member, or God alone. (It is conceivable but highly unlikely in practice, that a Catholic hermit could profess vows to a bishop, although this would not be recognized by Church law as it would be when professing via CL603.)
The Catholic hermits who profess their vows to another (or others) who is not the diocesan bishop acting in accord with the proviso of CL603 are in practice, but are not bound by church law, under the direction of their abbot, abbess, priest, or spiritual director and who oversees their proper, eremitic program of living.
To recapitulate, the distinguishing difference in the unofficially coined terms "private profession" and "public profession" of vows is as follows. The former profession of vows are not required by Church law to be professed in the hands of the hermit's local ordinary, or bishop, while the latter profession of vows is thus required, and thus becomes a public record per church law.
Any Catholic hermit may have his or her vows personally reviewed and approved by his or her or other bishop, but this does not relegate the hermit or the bishop to the stipulations of CL603. They would not be a matter of public Church record. Of course, all Catholics are, by nature of the Sacrament of Confirmation, to be in obedience to his or her bishop who acts as our hierarchical, local ordinary of the Catholic Church.
Thus, a Catholic hermit who professes vows (which must include the three Evangelical Counsels) per the stated conditions of the Church for eremitic consecrated life, but who do not profess the vows by the proviso of CL603, could conceivably make the vows "public" in the quintessential, populous meaning of "to make public".
However, to do so would obfuscate the reason most Catholic hermits have for centuries--historically and traditionally--professed their vows without publicity. They tend t0 avoid revealing their given name identities, location, or making "public" such as in parishes or community, their hermit vocations in order to strive to live a more hidden life.
For hermits who profess their vows per CL603, the desire may be to live their hermit vocations more anonymously. Retaining anonymity as a hermit is in such cases not so possible in the onset of profession in accord with CL603. There may be a Mass to which friends, family, or sometimes the parish are invited. There may be an article in the diocese newspaper by which the diocesan bishop recognizes the hermit whose vows he has received.
The hermit--and the diocesan bishop who has in accordance with Canon Law 603 recognized as [church] law, the hermit's professed vows--effectuates the hermit's profession as a matter of public Church record. After the initial mention or recognition in such cases as CL603 stipulates, most of these hermits in practice strive to live their program of life in a hidden manner.