We revisit the canonical problem of strategic facility location and study the power and limitations of randomization in the design of truthful mechanisms augmented with machine-learned predictions. In the strategic facility location problem, a set of agents are asked to report their locations in some metric space and the goal is to use these reported locations to determine where to open a new facility, aiming to optimize some aggregate measure of distance of the agents from that facility. However, the agents are strategic and can misreport their locations if this may lead to a facility location choice that they prefer. The goal is to design truthful mechanisms, which ensure the agents cannot benefit by misreporting. A lot of prior work has studied this problem from a worst-case perspective, but a recent line of work proposed a framework to refine these results when the designer is provided with some, possible incorrect, predictions regarding the agents' true locations. The goal is to simultaneously provide strong consistency guarantees (i.e., guarantees when the predictions provided to the mechanism are correct) and near-optimal robustness guarantees (i.e., guarantees that hold irrespective of how inaccurate the predictions may be). In this work we focus on the power of randomization in this problem and analyze the best approximation guarantees achievable with respect to the egalitarian social cost measure for one- and two-dimensional Euclidean spaces.