If you transition from the military and remain overseas, do you require sponsorship to work in the country you reside?

It is different in every country. Which one? In all of the Schengen countries (EU plus), you have to get a residency permit, which in most cases requires a work permit (not easy for an American, unless married to a local national). In many Central American countries, there are programs to encourage Americans to settle there (like tax breaks). A quick google search on the specific country should give you all the basic answers and then you can do more focused research.

Joe, I've lived and worked in Kuwait for 25 years now. If you have questions about here, or most anywhere in the Middle East, let me know.

Depends on the country. But most countries and the military will not allow you to separate overseas. There may be exceptions for spousal relationships when the individual is married to a native of another country and resides in that country. The military for the most part will not separate individual overseas (except in U.S. territories). Another issue to consider is that a majority of countries will not allow a retired U.S. military member to seek "direct" employment in their countries. And US orgs. overseas, often can only bring a limited number of people from the US for limited periods of employment.

Research each country. Obviously individuals can return as civilian employees of DOD contractors if the country doesn't challenge the employment. A person can seek a visa with a foreign country and relocate to live in that country with the permission of that government (unless on the restricted country list or having had access to restricted classified material/info.).

Numerous ex-pats live without "earning" a living in those foreign countries. For example, I met hundreds of them in the Republic of the Philippines. All those I met were either single or married to a native, except for a few who were involved in ministry work. Finally, once on retirement, Uncle Sam never forget you, your Federal taxes, or where you are located as long as you are drawing your retirement pay and/ or eventually social security.