For Amazon cross-border e-commerce sales, whether it is necessary to register your own trademark in each country depends on the individual or company's strategy and considerations.

Here are some common situations and suggestions:

1. International registered trademark: If you plan to conduct cross-border e-commerce business in multiple countries and want to protect your brand globally, you can consider international trademark registration. For example, register through the Madrid International Trademark System, which allows trademark applications to be filed in multiple countries or regions.

2. Regional trademark registration: If you only plan to conduct cross-border sales in a specific region or a few countries, you can consider regional trademark registration. This can protect your brand within the legal scope of these markets.

3. Amazon Brand Registry: Selling products on the Amazon platform usually requires brand registration and certification. Amazon's brand registration requirements may vary from country to country and platform to platform, and you need to register the corresponding trademark according to the regulations of the specific market.

It should be noted that the trademark registration procedures, requirements and fees may be different in each country. Therefore, before deciding whether to register a trademark in a particular country, it is recommended to consult a professional intellectual property lawyer or a professional trademark agency to understand the local legal requirements and make appropriate decisions.

In addition, trademark registration is only one way to protect your brand. You can also consider other intellectual property protections, such as patents, copyrights, etc., to ensure that your brand is fully protected in cross-border e-commerce.