There are two issues to discuss regarding the use of a goy shel Shabbat. One is menschlichkeit. You are apparently talking about a person who is paid by the community, whose main concern is its members’ most pressing needs. This includes taking people to the hospital, preventing large losses, and helping with the acute needs of individuals and groups. In some cases, the goy shel Shabbat is paid with the assumption that he will only be called upon to help for a relatively small number of acute needs. He might legitimately ask for more money if he is bombarded incessantly with requests to deal with small inconveniences. More importantly, he cannot be in two places at the same time, so if he is taking care of one person’s small need, he will be temporarily unavailable for another’s more significant needs, and that delay is sometimes crucial. (Regarding an “unofficial” Shabbos goy, there is a concern of not abusing his goodwill, but this concern can be eliminated or mitigated through pay or presents.)
Now we will discuss the laws of Shabbat. While it is not a simple matter, the standard halachic assumption is that in the case of some types of hints, it is considered as if the Jew did not make a request. The source for this assumption is the Magen Avraham,[2] who distinguishes between types of hints to resolve an apparent contradiction between halachic rulings. The Rama[3] says that whenever one may not do something on Shabbat, he may not hint to a non-Jew to do it for him. In contrast, classical sources indicate that one may tell a non-Jew that he cannot read a letter, thereby hinting to open the letter. The Magen Avraham answers that one is allowed to mention a need, but he may not tell a non-Jew to do an action in a way that will serve as a hint to do that which he wants the non-Jew to do. The Mishna Berura[4] and contemporary poskim accept this Magen Avraham.
However, there are two problems that must limit use of this leniency. First, when one sees a non-Jew doing forbidden work on his own initiative in a Jew’s home and/or using a Jew’s property on the Jew’s behalf, he is required to protest the activity.[5] This is because the Jew appears to the observer to be enlisting the non-Jew’s help in a forbidden manner. If it is necessary to protest when a non-Jew initiated the work, how could it be permitted to hint to him to do it in the first place? The other problem is that if a non-Jew does forbidden work on a Jew’s behalf, even without his involvement or knowledge, the Jew may not benefit from the result until enough time transpires after Shabbat for that particular chore to have been done then.[6] Again, a hint is certainly no better than not having raised the request in any form, and it should therefore be forbidden to benefit from the result on Shabbat.
Because of these factors, the poskim limit the efficacy of hinting to a non-Jew on Shabbat to cases such as the following: 1) A situation in which the Jew could have continued the activity that the non-Jew helped him with without the latter’s action (e.g., eating in a room where there was already sufficient light to eat by, when the non-Jew put on another light to make it more pleasant).[7] 2) The nature of the benefit provided is one of removing disturbances or impediments, not one of providing something positive new.[8] (Examples include shutting a light and perhaps opening an envelope.[9]) 3) The Jew will receive benefit only after Shabbat.[10]
In cases of significant need, in which it is permitted to ask a non-Jew straight out to perform what would be a Rabbinic violation for a Jew,[11] it is also permitted to benefit from whatever work was done.[12] Therefore, if one was careful regarding menschlichkeit and thus the need is apparently great when he calls the goy shel Shabbat, there are many additional cases in which the benefit element is resolved as well. In any case, if one is not familiar enough with the halachot, he should ask a rabbi before going to the goy shel Shabbat. Many communities assist in the matter by having the goy shel Shabbat keep “an instruction manual” handy for those who seek his help.