Come on! This is a direct quotation from Acts, Chapter 2, verse 32. It is part of Peter's "Men of Israel" speech.
It obviously means "This Jesus whom God resurrected, of which fact we are all witnesses." The word whom is understood. It's grammatically correct.
Compare it with other translations.
King James Version: "This Jesus hath God raised up, whereof we all are witnesses. "
NIV: "God has raised this Jesus to life, and we are all witnesses of it. "
Jerusalem Bible: "God raised this man Jesus to life, and all of us are witnesses to that."
The New World Translation merely translates more or less directly from the Greek; the other translations fill it out a bit more to ensure that the correct meaning gets across.
There is no sense whatsoever in which it says "Jesus God" as part of his name, and Jesus never once said that about himself. The misunderstandings arise when a passage is taken out of context, or is heard by people unfamiliar with slightly old-fashioned grammar.
Hope that helps.
Edit: Sorry, Black Sheep, I was writing my reply while you were posting yours, and I was interrupted by a phone call in the middle too!