It is not clear why you have to use Airdrop to transfer Contact Card between your own devices (Work and Private). Simply use the same email account that hosts your Contacts on both devices. You are free to use additional email accounts too.
When it comes to storing contacts on an iPhone or any other smartphone, you'll discover that they are not directly stored on the device itself or the SIM card. Instead, modern smartphones utilize synchronization with various email accounts to manage and store contact information.
When you add and enable the Contacts feature on your iPhone, it provides you with the ability to sync your contacts with a specific email account. This email account can be from various providers such as Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other email client server that supports contact synchronization.