The sensitivity of wide-parameter-space searches for continuous gravitational waves is limited by computational cost. Recently it was shown that deep neural networks (DNNs) can perform all-sky searches directly on (single-detector) strain data [C. Dreissigacker, Phys. Rev. D 100, 044009 (2019)PRVDAQ2470-001010.1103/PhysRevD.100.044009], potentially providing a low-computing-cost search method that could lead to a better overall sensitivity. Here we expand on this study in two respects: (i) using (simulated) strain data from two detectors simultaneously, and (ii) training for directed (i.e., single sky-position) searches in addition to all-sky searches. For a data time span of T=105 s, the all-sky two-detector DNN is about 7% less sensitive (in amplitude h0) at low frequency (f=20 Hz), and about 51% less sensitive at high frequency (f=1000 Hz) compared to fully-coherent matched-filtering (using weave). In the directed case the sensitivity gap compared to matched-filtering ranges from about 7%-14% at f=20 Hz to about 37%-49% at f=1500 Hz. Furthermore we assess the DNN's ability to generalize in signal frequency, spin down and sky-position, and we test its robustness to realistic data conditions, namely gaps in the data and using real LIGO detector noise. We find that the DNN performance is not adversely affected by gaps in the test data or by using a relatively undisturbed band of LIGO detector data instead of Gaussian noise. However, when using a more disturbed LIGO band for the tests, the DNN's detection performance is substantially degraded due to the increase in false alarms, as expected. © 2020 authors.