THz communications is envisaged for wide bandwidth mobile communications eventually reaching data capacities exceeding 100 Gbit/s. The technology enabling compact chip-integrated transceivers with highly directive, steerable antennas is the key challenge at THz frequencies to overcome the very high free-space path losses and to support user mobility. In this article, we report on mobile and multi-user THz communications using a photonic THz transmitter chip featuring 1D beam steering for the first time. In the proposed approach, 1D THz beam steering is achieved by using a photodiode excited leaky-wave antenna (LWA) in the transmitter chip. The on-chip LWA allows to steer the directive THz beam from 6° to 39° within the upper WR3-band (0.28-0.33 THz). The antenna's directivity is 14 dBi which is further increased to 23 dBi using an additional hemicylindrical Teflon lens. The 3-dB beam width and coherence bandwidth of the fabricated THz transmitter chips with lens are 9° and 12 GHz, respectively. The proposed approach allows steering the THz beam via the beat frequency of an optical heterodyne system at a speed up to 28°/s. Without using a THz amplifier in the transmitter chip, a data rate of 24 Gbit/s is achieved for a single user for all beam directions and at short wireless distances up to 6 cm. The wireless distance is successfully increased to 32 cm for a lower data rate of 4 Gbit/s, still without using a transmitter amplifier. Also, multi-user THz communications and the overall capacity of the developed THz transmitter chip is studied revealing that up to 12 users could be supported together with a total wireless data capacity of 48 Gbit/s. Fully integrated 2D transmitter chips are expected to reach wireless distances of several meters without additional amplifiers.