If I paid some service fee to a person or a business in a foreign country, can I create and send a 1099 to that person or business resides in a foreign country? How?

Generally, you do not need to issue a 1099-NEC or 1099-MISC to a foreign contractor or business if they are not a U.S. taxpayer and their services were performed outside the U.S.. 

However, if the foreign provider performed services within the U.S., they may be subject to U.S. tax reporting requirements, and a 1099 form might be necessary.

The IRS requires that you provide documentation showing the expense was ordinary and necessary for your business. This can include receipts, invoices, bank statements, or written agreements.  Have this handy in case if the IRS asks for it at a later date if you claim contract labor expense in your return.