If you are already at Columbia, holding a visa status sponsored by Columbia that allows you to conduct research or teach, you are not allowed to accept additional employment with very limited exceptions (as listed below).
Employment authorization for J-1 research scholars is limited to the activity described on your DS-2019 form, except for occasional engagements outside Columbia such as a lecture, participation in a symposium or a short-term consultation. The activity must be directly related to the objectives of your J-1 Exchange Visitor program and not delay its completion.
These statuses are employer-sponsored and employer-specific. This means you may work only for Columbia and only for the job duties specified when your employment status was approved.
To be eligible to accept and be paid for additional work (if allowed by university policy for your position), another employer would have to file a “concurrent” petition with USCIS for your same visa status.
Lectures or consultations outside of Columbia
If you are invited to give a lecture or consultation outside of Columbia, you are not authorized and it is a violation of your status to receive remuneration, such as honoraria. The host may pay your expenses directly, but may not pay you directly.