Given regular grammars (each is either left or right linear), does exist word/string so it can be derived from all regular grammars?
Your problem is PSPACE-complete. Indeed, the easier problem of DFA intersection is already PSPACE-complete, see for example Descriptional and computational complexity of finite automata—A survey by Holzer and Kutrib. For comparable results from the point of view of exponential time algorithms, see Problems on Finite automata and the exponential time hypothesis by Fernau and Krebs.

It is well-known that regular grammars are essentially equivalent in power to NFAs (even considering description complexity), and for this reason your problem is essentially equivalent to the NFA intersection problem, in which you are given a collection of NFAs and have to decide whether the intersection of languages accepted by them is empty &mdash; this is the problem considered in the papers above.

The same problem for regular expressions is also PSPACE-complete, see for example Complexity of decision problems for simple
regular expressions by Martens, Neven and Schwentick, which considers restricted cases of the problem.