It is also important to remember that because they are private vows it does not mean that they are made in secret. Many persons and communities make private vows, but they are made in public. Private is a canonical character and it has no bearing on the place where the vow is made or on the obligation to fulfill the vow. The vow must be fulfilled under penalty of sin.
Vows also take different forms. For example, the Benedictines do not make vows of chastity, poverty and obedience. They make two public vows: obedience and stability. Poverty and chastity are in the Rule of St. Benedict that they vow to obey. The Benedictine vow of poverty means that the individual may not own property, but the abbey may own as much property as necessary to make the life of the monks possible, unlike the solemn vow of poverty made by Capuchins. It does not allow them to own any property as individuals and they may only own common property as authorized by the General Superior or the Holy See.
Some third orders make private vows, but make them in public. The reason for this is that these third orders were never founded as orders per se. Rather they are public associations of the faithful. The term “order” was incorrectly applied to them by the laity who could not tell the difference between an association of the faithful and an order, because the difference is usually juridical, not always external.
I am not sure which third orders were founded as orders and which are associations of the faithful. The Secular Franciscan Order was founded as Secular Religious Order. It was the intention of the founder that it be a religious order for secular men and women, including diocesan priests. Because they have a canonical status as a religious order, with its own rule, constitutions and dependence on the Holy See, their profession is public, not private. This may also be the case with the Lay Dominicans as they were founded before the Dominican Friars were founded. They were originally an order of nuns and nuns are part of the laity.
Whether a vow is public or private, it remains morally binding until it is dispensed, suspended or commuted.
The difference between the vows has more to do with jurisdiction and with application. For example a simple vow of poverty allows religious and secular men and women to own private property according to the norms that are specified in their constitutions, if they belong to a community or to their intent if they are made by an isolated individual. The vow of chastity may include celibacy or not. The vow of obedience may mean obedience to the institute under specific circumstances, total submission without questioning or personal opinions as is required of Franciscan Friars, Secular Franciscans, Poor Clares, many cloistered Carmelites and Missionaries of Charity, or simply obedience to the Gospel and the Universal Church, but not to a superior.
Before making vows, one must make sure that one undestands the parameters. They may have the same names: chastity, poverty and obedience, but have different obligations.
By the way, the promises of obedience and celibacy made by secular priests are in a different category. They are not considered vows of consecration. They are public, but they do not make a diocesan priest a consecrated Christian. These promises are prerequisites for ordination, but do not consecrate the individual to any specific way of life. The individual is not consecrated by these vows. They are vows to a certain action, not a certain way of life.