Where you need to file is based on the rules for each state your business exists in (always) and operates in (in this case, NY). So you must take a look at the requirements separately.
For NJ, you are required to file a copy with the state, but only for 1099s that you issued for more than $1,000 (unlike federal's $600 threshold) or if you withheld any state income tax (doesn't sound like that would apply here). If your 1099s had more than $1,000 in box 7, you should file copies with NJ. Learn more at the NJ 1099 form filing instructions here.
For NY, you are correct that they don't require form 1099-misc filings for paying workers or businesses. Software used to file your federal 1099s, can be used to quickly file copies with your state as well.
