It's important to take measures to protect the privacy of your Bitcoin transactions and sharing a new wallet address should be done through secure means. Using an anonymous Nostr npub over Mullvad VPN or Telegram encrypted DM can be good options for protecting your privacy, but it's important to evaluate each option carefully.

Using Nostr or a VPN can help to prevent your IP address from being tracked, while using an encrypted messaging service like Telegram ensures that the message is secure and can only be accessed by the intended recipient. However, any messaging platform or communication method carries its own risks and it may not guarantee full privacy since they are still subject to potential hacking attacks.

In general, you should avoid sending wallets addresses or other sensitive information over unsecured channels such as regular email or SMS messages. Whenever possible use secure messaging applications that enable end-to-end encryption like WhatsApp's end-to-end encryption feature. Also don't forget that wallet software varies widely with regards to tracking features so private browsing mode might benefit too!