There is an interesting discussion of Acts 20:28 in Bruce Metzger's "A Textual Commentary on the Greek New Testament". In the first place, there is some question whether Luke wrote "to shepherd the Church of God" or "to shepherd the Church of the Lord". The reading "the church of God" is contained in codex Sinaiticus and codex Vaticanus while the reading "the church of the Lord" is contained in p74, codex Alexandrinus, codex Bezae and codex Laudianus. The difference between the two readings (in the original language) only concerns a single letter, whether it was a theta or a kappa. So the New English Bible, for example, reads "the church of the Lord" with a footnote that some witnesses read "of God". If it was "the church of the Lord" there is not the theological issue of asserting that it is God's blood which was poured out, which is the heresy of patripassianism. But Metzger reports that the Committee considered "the church of God" to be the more difficult reading and so more likely to be original.
They then went on to consider the last clause which could be translated as "with his own blood" or as "with the blood of his Own". He says that this absolute use of "own" as a noun is found in Greek papyri as a term of endearment referring to near relatives, and so it is possible it is a title early Christians gave to Jesus, comparable to "the Beloved".
J.H. Moulton confirms this in his "Grammar of New Testament Greek", pp.90-91, where he states:
Before leaving [the discussion of the word "own"], something should be said about its use without a noun expressed. This occurs in John 1:11 ["He came to his own [home], but his own [people] did not take him in."]; 13:1 ["... Jesus having loved his own that were in the world ..."]; Acts 4:23 ["After being released they went to their own [people] ..."]; 24:23 ["... forbid no one of his own [people] to wait upon him"]. In the papyri we find the singular used thus as a term of endearment to near relations. In [The Expositor VI. iii, 277] I ventured to cite this as a possible encouragement to those (including B. Weiss) who would translate Acts 20:28 "the blood of one who was his own." Matthew 27:24, according to the text of codex Sinaiticus and the later authorities, will supply a parallel for the grammatical ambiguity: there as here we have to decide whether the second genitive is an adjective qualifying the first ["I am innocent of this blood ..."] or a noun dependent on it ["I am innocent of the blood of this [man] ..."].