What licenses/endorsements or type ratings do you need to fly a fighter jet?
In the US, the FAA's requirements for obtaining authorization to fly former ,military aircraft is included in 8900.1 Volume 5, Chapter 9, Section 2.  

It is quite lengthy and covers more than just former military aircraft. As others have pointed out, most of these aircraft do not have a Type Certificate. They normally have a Special Airworthiness Certificate in the category Experimental Aircraft  Exhibition. Requirements to fly the aircraft are spelled out in the aircraft's operating limitations which are issued with the Special Airworthiness Certificate.  

From 8900.1:  


  A.    Background. The FAA requires pilot authorizations to operate
  some aircraft in the experimental category, including some former
  military, all turbojet‑powered (for this section, all turbojet‑powered
  includes all turbofan‑powered), all rocket‑powered, and all large
  (over 12,500 pounds maximum gross takeoff weight (MGTOW)) aircraft,
  and any other aircraft requiring specific pilot skills. Examples of
  such models are the Northrop F‑5, Bell P‑63, MiG‑15, Ju‑52, Mi‑24, and
  the BD‑5J. The FAA also requires authorizations to act as PIC of large
  or turbojet‑powered aircraft undergoing type certification.  
  
  1)    The requirement for an FAA authorization to act as PIC of
  certain experimental aircraft is contained in the FAA‑issued aircraft
  operating limitations. Pilots are required to comply with the
  FAA‑issued operating limitations by 14 CFR part 91, § 91.9(a).
  
  2)    Because these aircraft are not type certificated (TC), a type
  rating is not available. In the absence of type ratings for these
  aircraft, it is the FAA’s objective to ensure, for the pilots flying
  these aircraft, a level of safety and proficiency similar to what is
  available for an aircraft with a type rating.


The specific pilot requirements are also spelled out in the document:  


  A.    Former Military Turbojet/Turboshaft‑Powered Aircraft. To be
  eligible for an authorization to act as PIC of a former military
  turbojet/turboshaft‑powered aircraft, an applicant must:
  
  1)    Possess at least a U.S. private pilot certificate with an
  appropriate category and class rating for the configuration of the
  aircraft;
  
  2)    Hold an instrument rating;
  
  3)    Possess at least a valid U.S. third‑class medical certificate or
  equivalent (U.S. Military Flight Medical or U.S. driver’s license in
  accordance with the FAA BasicMed process (pilots using BasicMed may
  not exceed 250 knots indicated airspeed (KIAS)));
  
  4)    Have logged a minimum of 500 hours of pilot flight time in the
  aircraft category and have completed the U.S. armed services
  qualification checkout described in this section; or have logged a
  minimum of 1,000 hours pilot flight time, including 500 hours as PIC
  in the aircraft category, and have completed the training requirements
  of this section; and
  
  5)    If the aircraft is capable of supersonic flight, have a minimum
  of 250 hours of pilot flight time as PIC of a fixed‑wing
  turbojet‑powered aircraft, in a Group V, VI, or VII aircraft (see
  Figure 5‑173), or present proof of completion of a U.S. military
  qualification in a supersonic turbojet‑powered aircraft.
  
  B.    Former Military Propeller‑Driven Airplane. To be eligible to
  serve as PIC of a former military propeller‑driven airplane that has a
  MGTOW exceeding 12,500 pounds, or which has a horsepower rating of
  more than 800 horsepower and a VNE that exceeds 250 knots, an
  applicant must:
  
  1)    Possess at least a U.S. private pilot certificate with an
  appropriate category and class rating;
  
  2)    Possess at least a valid U.S. third‑class medical certificate or
  equivalent (U.S. Military Flight Medical or U.S. driver’s license in
  accordance with the FAA BasicMed process (pilots using BasicMed may
  not operate an aircraft with a MGTOW above 6,000 pounds));
  
  3)    Have logged a minimum of 500 hours of pilot flight time; and
  
  4)    Have completed the training requirements of this section.


That last item can be a challenge as it requires finding someone with the proper authorization to provide the training.  

More information is available at the FAA Vintage &amp; Experimental Aircraft Program web page.