I answered a similar question here.

More specifically the practice of assigning jobs by distribution mostly applied to college or university graduates. I’m basing my answer on the relevant legislation issued by the Soviet Ministry of Higher Education in 1968, but it was based on similar legislation that dated back to 1933 at the very latest.

Usually the institutions would have a special commission that would assign the graduating class during the last several months (no later than 4) of their program, usually based on the requirements of the relevant industries or sectors.

For example a factory or company could issue a request to either the regional department of education or labour, or a specific university, asking for a certain number of “young specialists” to fill their vacancies, usually outlining the responsibilities, pay, benefits (primarily housing), if any, and other preferences such as gender (though almost nobody would specifically spell it out as a requirement) or fitness.

The commission was composed of heads of faculties and/or programs at the educational institution, representatives from the relevant employer sectors, and was required to be chaired by the university rector. It was supposed to match the graduates with the labour requests, taking into account the academic standing, grades, family status, and health, as well as other factors, including personal preferences. If you were deemed to be a suitable candidate for more than one vacancy (most commonly based on your grades), you would normally be allowed to choose the placement you preferred.

The decision was supposed to be taken by commission members through an open vote. And both the graduate and the potential employer were to be given the opportunity to object to commission’s assignments.

That said, the graduate didn’t really have a final say in their job assignment, though trying to influence the commission’s decisions in order to secure a better placement was not altogether unheard of. One couldn’t, however, refuse a job assignment outright if one didn’t like it, as it was considered part of one’s civic duty, in return for free education if you will (the Soviet Constitution enshrined work not only as a right, but also a duty), though any objections were supposed to be taken into account.

Depending on the period and the relevant legislation there was a whole slew of specific accommodations based on one’s status. For example in the 1968 version of the act, the commission was required to accommodate graduates with certain disabilities, or disabled parents, pregnancy or the presence of children under the age of one, marriage to someone serving in the armed forces, or graduating alongside a spouse, mostly by assigning them to a position in their area of residence.

Aside from that, once you “fulfill” your job assignment (which was usually 3 years), you were free to seek employment elsewhere, but you could not quit until the mandatory period elapsed (the employer also couldn’t fire you or reassign you to a position that is not related to the one you were originally assigned by the commission).

You can find the texts of the relevant acts and orders that I referred to from 1933, 1946, and 1968 in Russian here, here, and here.