No notching is allowed in the middle third. As a licensed home inspector, that’s all I really need to say.
Sistering is an interesting approach, but I’m not convinced that it overcomes the problem created by the -shall we say- extreme notching in the middle third.
I’m not citing any code. See p. 70, “Essentials of Home Inspection-Home Reference Book,” Carson-Dunlop & Associates Ltd. c. 2003.
Again, not wishing to offer any engineering advice, I would like to reiterate the following comments about joist strength. (quoting ibid, p. 27)
“It is interesting to note that the strength of a joist comes largely from its depth. Doubling a joist by putting another of the same size beside it will double its resistance to bending. Doubling the depth of a joist increases its deflection resistance eight times. For example, the equivalent strength… of two 2 x 10s compared to four 2 x 8s can be compared by multiplying the base times the height cubed. It is interesting to note that two 2x10s are more resistant to deflection than four 2x8s for conventional house framing purposes.”
The point being, depth trumps thickness. If you do the math (base times height cubed, actual measurements) you might be surprised.