  We address the problem of goodness-of-fit testing for H\"older continuous
densities under local differential privacy constraints. We study minimax
separation rates when only non-interactive privacy mechanisms are allowed to be
used and when both non-interactive and sequentially interactive can be used for
privatisation. We propose privacy mechanisms and associated testing procedures
whose analysis enables us to obtain upper bounds on the minimax rates. These
results are complemented with lower bounds. By comparing these bounds, we show
that the proposed privacy mechanisms and tests are optimal up to at most a
logarithmic factor for several choices of $f_0$ including densities from
uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In
particular, we observe that the results are deteriorated in the private setting
compared to the non-private one. Moreover, we show that sequentially
interactive mechanisms improve upon the results obtained when considering only
non-interactive privacy mechanisms.
