It is good that you are sensitive to both the ecological elements of recycling and the halachic propriety of what you take from where. This question requires research in two areas: monetary Halacha and the attitudes of the people in charge of the recycling effort.
We will start with Halacha. When the bottles are placed in the receptacles, does the recycling company or authority acquire them immediately, which would make it necessary to receive the explicit or implicit permission of the owner? Or is it possible for you to take the bottles before any employee acquires them on behalf of the authority?
A kinyan[1] via chatzer, i.e., when an object is on the property of the acquirer, has a rare quality – it can work without the involvement, presence, or knowledge of the acquiring party.[2] A variation of kinyan chatzer is a kinyan through a kli (utensil); if an object is placed in someone’s utensil, he acquires that object.[3] While this does not usually work if the kli is in the public domain, that is because people do not have the right to leave their personal utensils in the public domain to use as they like. However, when they have permission to keep their kli in a given area, the kinyan works even in the public domain.[4] Recycling receptacles are certainly put out with permission and thus fall into this category.
Another requirement for kinyan chatzer is that if the acquirer is not present, the kli must be located in a place that is guarded on behalf of the acquirer.[5] While there are various explanations for this halacha, the basic idea is that in the acquirer’s absence, the chatzer has to serve as a shaliach (agent) of sorts, and to fit the role, it must be a reliable guarantor that the object will not be taken by any passerby.[6] On the other hand, many authorities maintain that if someone is giving the object to the acquirer (as opposed to a case in which a lost or un-owned article finds its way into the chatzer), it is sufficient that the giver is watching it at the time that he puts it in the utensil.[7] Furthermore, the Netivot HaMishpat[8] says that if the kinyan is done in a utensil with walls, there is no further need for it to be guarded.
Based on what we have learned, the recycling authorities acquire the bottles that are put in the receptacles. Therefore, we need to determine whether the people in charge of the recycling efforts allow you to take the bottles. (In truth, even if there was a halachic deficiency in the recycling authority’s kinyan, it would be best not to take the bottles against their will, but it is worthwhile to know the halacha for cases regarding which you are not confident about what they would say.)
The most important research conducted on many “halachic” questions is determining the simple facts. I contacted the recycling authority, and I was told that at present (I cannot tell you about the past or the future), the Jerusalem municipality operates the collection efforts and owns the recycling receptacles. The municipality workers in charge of the collection told me that the Jerusalem municipality is not bothered at all if people take bottles and reuse them. In general, my research indicates, recycling is not particularly financially rewarding these days; it is done primarily to alleviate the need for landfills or pollution through incineration. Thus, when someone takes the “usable garbage” home, it does not at all harm the national recycling efforts. One of the officials thought that it would be educational to have the children bring bottles themselves, but we leave that to your educational and logistical wisdom.
We do have a slight concern that, in certain settings, some people might view you as “raiding” the recycling bin, and thus there could be a chillul Hashem entailed. However, in most cases, that is not necessarily true. We cannot give an absolute assurance that in other municipalities or in other times the situation will be the same, but in Jerusalem of the foreseeable future, you may take the bottles without fear of stealing.