This is a great question. There are several challenges in doing good and responsible psychology. Let me list three main problems.
First, unfortunately we live in a world that needs money for research. So, I would say that one of the major challenges in doing research is to get grants to carry out research. Without financial backing, no research can go on. Moreover, research costs can be very high.
Second, when you are dealing with psychology, you are dealing with the human personality. For this reason, there are challenges of complexity. The human personality is deep and theories can change and get overturned. In other words, the subject matter itself poses a challenge.
Third, it is often times hard to get good subjects to be in an experiment and to experiment ethically. Think about it. Who would want to be in an experiment to be studied, and how are ethics maintained?