On occasion I will travel on my company's corporate jet. From a legal/FAA standpoint, are there any regulations against receiving flight training from a CFI with the proper ratings and currency during those flights? I'm a private ASEL pilot, so this would be working on an instrument/multiengine/type rating. And does the answer change if the jet is certified for a single crew member vs requiring 2 pilots? Assume that my employer is ok with this if it's legal.
It is an interesting question and needs quite a few FAR's to figure out. Let's start with having other people on board:
If the aircraft is not certified for single-pilot operations, then you cannot based on FAR 61.89(b)
(b) A student pilot may not act as a required pilot flight crewmember on any aircraft for which more than one pilot is required by the type certificate of the aircraft or regulations under which the flight is conducted, except when receiving flight training from an authorized instructor on board an airship, and no person other than a required flight crewmember is carried on the aircraft.
If it is certified for single-pilot operations, you may be able to receive training as long as you are not PIC and you are not sitting in the left seat (as almost all single-pilot jets that are certified single-pilot usually have a restriction to left seat only).
However the real rub comes in when you consider that you are a private pilot, not a commercial pilot, and the FAA's extremely broad definition of "compensation". On more than one occasion, the FAA has held (and won in court) that getting flight time in furtherance of a certificate is compensation. Technically just receiving your normal compensation (hourly wages) is enough to meet the definition of being compensated for flying.
The only way to get a real good answer to this, because it is a very broad subject and open to legal interpretation, is to engage your local FSDO.
I'm guessing though that the ultimate answer will be no. Aside from the issues with the FAA and FAR's depending on what operations your employer is conducting under (121, 135, 91, etc), your employer will also have to consider aircraft insurance. Many corporate jets are insured only for certain pilots or pilots who meet minimum qualifications and the insurance policy may completely bar primary training of any kind in the aircraft.
So talk to your FSDO, if you get an OK from them, the next step is to talk to the corporate attorney or an aviation lawyer and have them review the insurance documentation to see if you are allowed to sit in one of the seats and manipulate the controls.