Yes, the slaves had almost no rights. (They only could not be killed). But that does not mean they were forbidden to do anything or to be active. That only means that they could be punished for their actions postfactum. Or not punished. Or even prized.
The Rome Empire/Republic was a living, and effective structure. And it definitely was not a rigid monolyth, trying to regulate everything. Don't forget, slaves often became so rich personally, that they could buy themselves out. So, they could have a great allowance for initiative.
And the system was very diverse. The slaves were subjects not to the Law of province and state only, but also to the rules of the owner and his family and to the traditions. Also, every latifundia had some organizational structure, and the slaves had their roles in it.
So, the answer depends on the rules and roles set for you. If you were a cleaner, the missing owner can interest you only in the sense what rooms to clean today and what tomorrow. But if you were a head of "guards", or some other safety structure, it would be your direct business to go and try to find and help your master. If you were the part of the economic government of the latifundia, and your actions need some official acknowledgment of the master, of course, it is up to you to know where he is in order to get it. And if he is missed, you should look for him or organize appropriate actions.
And of course, if you are an only slave of the owner, you simply must find the missed master - you would be expected to do that.
But you were not LEGALLY required to do it. It was not due to law, but due to your duties in the estate you belonged to. And due to your personal human qualities, too, of course. The Romans understood good and evil.