Never.
Sub-contractor? Someone you are paying? They will get your tax number when you give them a 1099-MISC, they don't need it now and they would never need a copy of your SS card.
If someone is planning on paying you more than $600 for subcontracting work, they need to issue a 1099-MISC form to you, and they use the form W-9 to request your taxpayer number. Form W-9 uses a signature certification. There is no requirement to even show your SS card and you should never let anyone take a copy of it.
If the person is doubting your authorization to work in the US, they would use form I-9 to collect proof that you are authorized to work in the US. Such proof can include an SS card, passport, birth certificate or other documents. But the person requesting the I-9 does not get copies of the documents. You show the documents to the person completing the form and then both of you sign it that you have seen the documents, but they don't keep copies.
If you are a contractor or small business person you may want to get an EIN from the IRS to use instead of your personal SSN. It can be done online and takes less than 20 minutes.
https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n...
