It is my understanding that the amount on a 1099 corresponds to the money received (by the recipient) in that calendar year. In the case of a missionary, I would understand this to mean "the last/final check your church gives YOU for at the end of the year." This is always the way it has been handled (with my donations) in the past and should there be a discrepancy between the monthly checks (received by the missionary) and what the church is reported.

Simplified: If the church decides to hold donations sent at the end of December and give you the money in mid- January is that still considered income YOU received in the prior year?

When they received the money isn't relevant to you or your 1099. If you are paid from 1/1/11 on, it is income this year. But if they write checks at the end of December and mail them with 2010 postmarks, then they count for 2010, regardless of when they arrive in the mail to you.
You might want to clear this up with the church to see what they are claiming on their tax return so that there is no discrepancy (and try to correct it if it's wrong).