Self-endowed chair in the US and Europe Endowed professorship is very common, but the financial endowment normally comes from external sources. Is it also common that a professor/research endow his/her position to establish a research group in the target university?

It should be an excellent approach for mid-level professors/researchers who can establish a research group in a top university (where it is normally hard to get that position) and get research fund from external sources such as industry and funding agencies.

This is of mutual benefit for both the university and professor, but I have not heard about such positions. Is it uncommon or just through private contracts?

UPDATE: Apparently, I confused endowed and named chair with research professor. I asked the question about research professor here. However, I still do not see a contradiction between research professor and self-endowed professor. Does a endowed chair must be necessary result of a huge donation to university? or a foundation can merely fund a professor position. If it is the latter case, one should be able to fund his own endowed position too.
Endowed professorship typically are the result of a large donation of money to the university that allows them to fully fund a position in perpetuity based on the earnings on the initial donation. This is generally much larger than what an individual can bring in from external sources. If you manage to bring in that type of income, you will likely have little problem obtaining a permanent position.

I am aware of one independently wealthy individual who made a large enough donation to establish his own professorship. Basically he spent a small fraction of his wealth (which would be a large fraction of the wealth of most academics) to become a full professor with minimal teaching and service duties. His research was funded from his charitable foundation and I believe the foundation paid the standard overhead rates. He was reasonably productive throughout his entire career and probably could have obtained a professorship without using his personal wealth, but given the competitive nature of the job market and limited availability of positions, probably not at the institution of his choosing.