What effect do the two speakers have on the reader's picture of "The Wanderer"?
The effect the two speakers in "The Wanderer" have on a reader depends largely on the sophistication of the reader.
Any text written and copied by hand before the advent of the printing press is a candidate for interpolations. This Anglo-Saxon poem as we see it today, told orally for generations before it was ever written down, is a prime candidate.
In other words, the narrator is probably a character added by a Christian monk who wrote the oral poem down for the first time, or who copied the poem after some other monk had written it down. The writer or copier "added his two cents worth," so to speak. The lines attributed to the narrator are probably lines added to Christianize the text. And this text happens to be the text that survived.
Notice that the rest of the poem--the Wanderer's own words--are "pagan," not Christian. They deal with the loss of a way of life and a man's reaction to the loss of that way of life. The only mention of God's pity, etc., comes from the narrator. The Wanderer himself battles against fate, and says nothing about benefiting from God's pity. His way of life has been devastated and destroyed (Anglo-Saxon life was notoriously unstable), and he is feeling despair.
No copyright laws existed in Anglo-Saxon England, of course, and the concept of private ownership of a text wasn't the same then as it is today. There wasn't anything unethical about a monk Christianizing a text. In fact, we, today, owe the monks a great deal of gratitude. If not for what appears to be their open-mindedness and interest in local literature and traditions, we would have few texts from the period.
Of course, an interpolation-creating monk could have just been trying to convert the "pagans" by linking traditional poetry to their Christian message, but whatever the motivation, Anglo-Saxon literature survives, in part, thanks to well-educated monks.
The narrator, then, probably doesn't belong, and, thus, you have only one speaker to worry about, rather than two.