What screening process do regional centers conduct on potential EB-5 investors?

They will want basic personal information and to review your source of investment funds. Of course, your immigration attorney will ultimately prepare and file the authenticated source of funds with the I-526 petition. Regional centers will also want to determine if you can be considered an "accredited investor," which requires information on your net worth or earnings. Beyond that, some regional centers or projects will want to schedule some kind of meeting (in person, telephone, video chat) to meet you and discuss your goals and their project.

Regional centers review your financial profile to make sure you are a qualified investor. I would say this is your attorney's responsibility to question you fully about your background to make sure there are no grounds for inadmissibility, so that neither you nor your attorney are there for a big surprise.

Regional centers generally will ask potential EB-5 investors to complete an Investor Suitability Questionnaire asking for specific information to determine whether or not you are an accredited investor. Some regional centers require more information and ask to document source of funds in more detail than the others. No background checks are usually conducted. Your immigration attorney will generally discuss with you issues of inadmissibility if any may be applicable in your case.

They have to show you are an accredited investor. USCIS will do background checks on you.

They will do a background check to save you from being eventually disqualified after you have made an investment. They want to know if you have a criminal record, if you have been to the United States and were removed or overstayed your visa, worked here without authorization, etc. They will also seek to know the sources of your funds.