Experiments have definitively demonstrated entanglement, and ruled out any kind of slower-than-light communication between two separated objects. The standard explanation for this behavior involves what’s called nonlocality: the idea that the two objects are actually still a single quantum system, even though they may be far apart. That idea is uncomfortable to many people (including most famously Albert Einstein), but it preserves the principle of relativity, which states in part that no information can travel faster than light.
To get around nonlocality, several ideas have been proposed over the decades. Many of these fall into the category of hidden variables, wherein quantum systems have physical properties (beyond the standard quantities like position, momentum, and spin) that are not directly accessible to experiment. In entangled systems, the hidden variables could be responsible for transferring state information from one particle to the other, producing measurements that appear coordinated. Since these hidden variables are not accessible to experimenters, they can’t be used for communication. Relativity is preserved.