This is actually a question with an interesting caveat and one whose answer depends very much on whether you’re talking about the Antique or Medieval world, because between the two lies a vast difference in educational practice and because of that the people who can write and read (or just the one).

For the Imperial and Late Antique world, for scribes, the answer is yes, you had to know how to read - and to learn things like multiplication and fractions, and all the systems of weights and measures one might need on the job - but interestingly enough for lower-level bureaucrats the answer is solidly “kind of” and in Late Antique education, there is substantial evidence that the ability to write was taught before the ability to read, meaning that anyone with some but only a very little education may have had some rudimentary copyist skills but still not have been able to read. Cribiore calls this group “slow writers”: able to sign their own names, as this was a practical and early-learned skill (we have ostraca written by very beginner students with only the early stages of the alphabet which have names in the same hand as the letters being learned, indicating that this was perhaps one of the first things learned, at least for some), and able to copy at least a little.

As for being able to get away with not knowing how to read but knowing how to write, we have records of a village clerk in the Egyptian town of Ptolemais Hormou in the late 2nd century CE who was able to sign papers with “Petaus, village clerk, [I] signed this”, copying the phrase from a model when he needed to use it. Just his name, his title, and the single verb he could use for everything. Petaus used practice sheets to work on this particular phrase, since it was the one he needed to actually submit documents. Harris suggests the possibility that Petaus could have been able to read Greek, just not to write it (Harris 278) but since copying skills were learned before reading this is unlikely. Cribiore also points out that Petaus was unable to correct once he made a mistake, at the very least heavily suggesting that he wasn’t able to read and see the errors (Cribiore 172). And since Petaus himself was called upon to look into the case of a different village clerk who was accused of illiteracy, he was hardly the only one, just the one whose story we have.

If you're at all interested in Roman education (and how we can pick out writing and models used for educational purposes and the stages learners went through), I highly recommend giving Cribiore a read.

Cribiore, Raffaella. Gymnastics of the Mind: Greek Education in Hellenistic and Roman Egypt. Princeton: Princeton University Press, 2001.

Harris, William V. Ancient Literacy. Cambridge: Harvard University Press, 1989.