Beam alignment is an important task for millimeter-wave (mmWave) communication, because constructing aligned narrow beams both at the transmitter (Tx) and the receiver (Rx) is crucial in terms of compensating the significant path loss in very high-frequency bands. However, beam alignment is also a highly nontrivial task because large antenna arrays typically have a limited number of radio-frequency chains, allowing only low-dimensional measurements of the high-dimensional channel. This paper considers a two-sided beam alignment problem based on an alternating ping-pong pilot scheme between Tx and Rx over multiple rounds without explicit feedback. We propose a deep active sensing framework in which two long short-term memory (LSTM) based neural networks are employed to learn the adaptive sensing strategies (i.e., measurement vectors) and to produce the final aligned beamformers at both sides. In the proposed ping-pong protocol, the Tx and the Rx alternately send pilots so that both sides can leverage local observations to sequentially design their respective sensing and data transmission beamformers. The proposed strategy can be extended to scenarios with a reconfigurable intelligent surface (RIS) for designing, in addition, the reflection coefficients at the RIS for both sensing and communications. Numerical experiments demonstrate significant and interpretable performance improvement. The proposed strategy works well even for the challenging multipath channel environments.