While the proto-writing systems are based on pictograms, both the Sumenrian cuneiforms and the Egyptian hieroglyphic (used for inscriptions on stone) and hieratic (use for writing with a reed brush on papyrus) have made the transition towards having phonetic symbols, used together with ideograms.
Using the phonetic sign subsets of the Egyptian and Sumerian scripts, it was possible to write any sentence that could be spoken in their languages.
This was the most important advance in writing and both in Mesopotamia and in Egypt there is evidence about this transition from an earlier writing system that could write only a subset of the words of a language, so it could not be used to write arbitrary sentences, but only things like lists of objects with their amounts and owners, like needed for accounting, to a writing system that added phonetic symbols for writing any words that did not have their own symbol.
I cannot read the paywalled article, but it seems that now there is evidence that also the Proto-Elamite writing system has also passed around the same time through this transition from having only symbols for certain words to having phonetic symbols too, e.g. for syllables, which can be used to write arbitrary words and sentences.
Before phonetic symbols began to be used, we cannot know the language spoken by the users of a proto-writing system.
While in Egypt there is little doubt that the first users of writing spoke some kind of Old Egyptian, in Mesopotamia there is doubt the users of the first proto-cuneiform writing system spoke Sumerian. However, by the time when phonetic cuneiform signs were introduced, the language of the writers was Sumerian.
In the territory later known as Elam (in the West of present Iran), it is not known what language was spoken by the users of the Proto-Elamite writing system. It could have been an ancestor of the Elamite language spoken a millennium later, or it could have been a completely different language. Elamite is not related to the Indo-European languages that spread much later in that territory, like Old Persian.