Receiving s’char Shabbat is indeed Rabbinically forbidden,[1] like commercial activities in general, lest one come to write.[2]
The most common method that allows one to receive money for work that was done on Shabbat is through havla’ah. This means having the Shabbat-related earnings “swallowed up” by combining them with weekday pay for a period of employment that happens to include Shabbat.[3] You apparently assume that the applicability of havla’ah depends on the payment period. In other words, each payment has to include pay for work not related to Shabbat or Yom Tov. Therefore, you would forbid a paycheck for a payment period (e.g., month) in which there is no weekday work.
However, poskim point out that “havla’ah units” are determined not by the period of payment but by the period of employment.[4] The period of employment is the time during which there is a commitment to continue the employer-employee relationship, without the ability to back out under normal circumstances. This has ramifications both for leniency and for stringency, depending on the particular situation. If the employee is owed for work on Shabbat and the employer is not obligated to continue the employment during a period that includes weekdays, the work on Shabbat is viewed independently and the worker is forbidden to receive the pay. One common example is a babysitter, who usually gets hired for each job individually. Therefore, getting paid at one time for separate babysitting jobs, one during Shabbat and other(s) during the week, is forbidden.[5] Your situation, in contrast, is in all likelihood an example of the lenient ramification. A shul usually hires youth workers for “a year” (often Sept.-June), which is the relevant time unit even if the payments are made in monthly installments. If that is the case, then since the year includes work on Tu B’Shvat and Purim, the pay is permitted, as your friends surmised.
There is often another, related leniency in the application of havla’ah. Some suggest[6] that the preparation that a worker does during the week (e.g., a chazan’s practice or a waiter’s setting up before Shabbat or cleaning up afterwards) justifies his receiving pay for his work on Shabbat and Yom Tov. However, for this to constitute havla’ah, it does not suffice for there to be a theoretical possibility of preparation or an insignificant amount thereof. It is necessary that there be obligatory work that is time-consuming enough to warrant pay.[7] In this vein, there is an assumption that youth workers, beyond their observable interactions with the children on Shabbat and Yom Tov, have necessary preparatory work that is slated to be done on a weekday. This can include buying prizes or food, setting or cleaning up, or preparing props. The shul can ensure from the outset that there will be significant weekday components of their jobs by requiring the leaders to come to a training session or a meeting or to call the children and/or parents with whom they will be working. One such significant responsibility during the employment period suffices.
The matter of chazanim introduces one more potential justification for those in your position to receive pay. There are two opinions in the Shulchan Aruch[8] regarding whether the prohibition on s’char Shabbat applies to mitzva activities. While the Shulchan Aruch seems to lean toward stringency, the Mishna Berura[9] acknowledges that the more prevalent minhag is to be lenient. Contemporary poskim leave this question open.[10] Whether or not a synagogue’s youth groups are considered mitzva activities depends on the content of the activities.