UTXO sharing with Bitcoin covenants like CheckTemplateVerify (CTV) can improve on-chain privacy in a few key ways:

1. Shared ownership of UTXOs: With CTV, multiple parties can jointly own a UTXO in a way where all their signatures are required to spend it. This makes it harder to trace ownership and spending to a single entity.

2. Obscured transaction graphs: When a UTXO is spent that had shared ownership via a covenant, it breaks deterministic links in the transaction graph. Outside observers can't easily connect the inputs and outputs, improving privacy.

3. Decoy inputs and outputs: CTV allows the creation of transactions with many decoy inputs and outputs not actually involved in the transfer of funds. This makes it much harder to determine the true sender, recipient, and amounts.

4. Concealing address reuse: Normally, reusing a Bitcoin address harms privacy by making it easier to link transactions. But with CTV, UTXOs can be spent back to the same covenant-controlled address without revealing that it's a reused address.

5. Privacy-enhancing protocols: CTV enables the trustless execution of privacy-focused smart contract protocols on Bitcoin like CoinSwaps, payment channels, atomic swaps, and certain forms of CoinJoins. These can leverage the UTXO-sharing capabilities of CTV.

So in summary, CTV allows a form of obfuscated, multi-party UTXO ownership that makes it very difficult to trace the flow of funds, link addresses, or determine true senders and recipients - providing a significant boost to on-chain privacy when used correctly in Bitcoin transactions and contracts. The privacy benefits compound as CTV adoption increases.