It would depend on the site.
I certainly have trusted Bonanza and Etsy in the past. Technically there should be no problem.
The problems that arise are:
Sales are very slow on other sites. It is important to constantly be removing any items sold on eBay from the other sites promptly in case you get a double sale of a single item. (And vice versa, but that's unusual).
Your eBay information may appear on the other site, for example, links to an eBay store. This isn't really a problem for you, but may confuse your customer.
Pricing. Bonanza in particular has a default that discounts your asking price by the difference in eBay and Bonanza fees. You can manually override this.
