If you click on the three dots icon in the top right of the screen, in any coin's Overview tab in Coinomi, you see an option to "Sweep Wallet". This is a very different process from "Importing" a private key.

When you're importing a private key, the coins are visible and accessible from both the source and the destination wallet (the wallet the key was exported from, and the wallet the key was imported into), and can be spent from either wallet. This is not supported by Coinomi.

When you're sweeping a wallet by entering the private key a transaction is created which empties the source wallet (the wallet the key was exported from) and sends all funds to the destination wallet (the wallet the key was swept into, in this case, your Coinomi wallet). The funds are visible and accessible only from the destination wallet.

The transaction and its details, including the small transaction fee needed - paid to the miners of each coin's network and NOT Coinomi - are presented to the user in the next screen. 

The reason Coinomi opts for sweeps instead of imports concerns the Restoration process. When you Restore your wallets in Coinomi using your Recovery Passphrase, any swept wallets would be recovered too, since the funds were transferred to your Coinomi and the source wallet was emptied. This would not be the case with imports.