Two protocols of quantum direct communication with authentication [Phys. Rev. A 73, 042305 (2006)] were recently proposed by Lee, Lim, and Yang. In this paper we will show that in the two protocols the authenticator Trent should be prevented from knowing the secret message. The first protocol can be eavesdropped on by Trent using the intercept-measure-resend attack, while the second protocol can be eavesdropped on by Trent using a simple single-qubit measurement. To fix these leaks, we revise the original versions of the protocols by using the Pauli Z operation σz instead of the original bit-flip operation X . As a consequence, the attacks we present can be prevented and accordingly the protocol securities are improved.