It is forbidden to transfer on shabbat between a reshut hayachid and a reshut harabim, or 4 amot in a reshut harabim. Additionally, the Rabbinical law of eruv forbids transferring from houses into an outer courtyard, or from a courtyard to the outer mavoi. But aside from this, is it generally permissible to transfer in between two reshuyot hayachid - e.g. from my courtyard to my neighbours'?
Building on what
Double AA and
Joel K have said: מדאורייתא this is definitely permitted. The decree of King Solomon declared multiple contiguous רשויות היחיד in which people live in defined, separate areas as "separated" in a sense, and said that they they need to be "combined" in order to allow carrying between them.
However there are some forms of רשות היחיד in which people don't live. For example, a raised platform that is 4x4 amot and 10 tefachim high within a רשות היחיד (e.g. a trampoline in a private garden) is a separate רשות, or an open area enclosed by houses but not set aside for use (קרקף שלא הוקף לדירה) so - could one carry between these (מדרבנן) without an eruv?
To my understanding this is a מחלוקת תנאים in the beginning of chapter כל הגגות in עירובין (89a). R' Meir holds it is forbidden, while R' Shimon holds it is permitted, and the halacha is like him. So according to R' Shimon, we can carry around the contiguous רשויות היחיד in a city, going from the courtyards to the flat roofs (which according to him aren't seen as part of the living area) to the surrounded open areas (קרפיפות) and so on. Naturally the objects cannot have begun shabbat in a private living area (in which case they need an eruv to be taken out of that area).
(Cases that imply that transfer between two courtyards is forbidden without an eruv, such as the window (Eruvin 76a) or the well (86a) between two courtyards, are talking about transfers to/from the private houses, or objects that began shabbat in a private house. See for example Aruch HaShulchan 376:1)
According to Torah law that is indeed permissible since it's not one of the two forbidden categories of transferring you mentioned. However, King Solomon enacted that it be forbidden to do so unless the neighbors first enact an Eruv between them, symbolically joining their distinct private domains into one, lest people get confused and come to transfer objects in a biblically forbidden way. (Rambam, Laws of Eruvin, Chapter 1)