The word "scan" tends to track with the English-language idea of the eyes being receptors. These eyes are looking throughout all the earth, gathering information. The Hebrew word shuwt means literally "to push forth," which simply implies "to go" or "to run." Thus, the King James version reads that the eyes "run" throughout the whole earth, which is a literal rendering. The New American Standard, the Revised Standard Version, and the New International Version use the word "range," while The New Living Translation reads "search."
The translation in the Keil & Delitzsch commentary uses the word "sweep," which seems to be a close synonym for "scan." However, when they explained it, they said that this word has the implication of influence—that their influence sweeps throughout the whole earth. If that is the case, if the messengers of the seven churches are the seven eyes, then these seven messengers have influence that runs over the whole planet.
Their influences is not localized, say, to Judea where the original prophecy was uttered and to which it was given. At the time of Zerubbabel, only a few thousand Jews and Levites were interested in this prophecy. Yet, he is saying that, in its fulfillment, the prophecy applies worldwide. It is not centered just on Jerusalem or just on a small area of Judea, but the influence of these seven eyes "pushes forth" throughout the whole earth.
This seems to fit what is happening today in God's church. Zechariah's prophecy does not say that their influence is necessarily strong throughout the whole earth, but it exists globally. This adds to the several things in this chapter that promote the idea that a worldwide work is being done. Even the last three words in this chapter speak of "the LORD of the whole earth." Zechariah 4 suggests quite strongly that this is a worldwide phenomenon.
Richard T. Ritenbaugh
The Two Witnesses (Part Five)