A bird's respiratory system is about 8x as powerful as ours, but not 8x as effecent.
- A bird's lungs are about 2x the proportional size of a terrestrial mammal's.
- Air sacs approximately double a bird's respiratory volume as compared to a terrestrial mammal's lungs making the total volume of their respiratory system about 4x that of a terrestrial mammal's.
- Birds typically have about 1/2 of their air sacs inside of pneumatic bones. This means that by virtue of their bones, they can reduce how much body cavity space they need for their respiratory system by 25%.
- While a bird's respiration is unidirectional thanks to its air sacs, this does not make their respiratory system inherently more efferent. What it does is normalizes the oxygen intake so that the bird can absorb oxygen on both the inhale and the exhale which means that it does not have to stop intaking oxygen to exhale. While this doubles the efficiency of the lungs, when you add in the extra volume the air sacs take up, it is kind of a moot point in terms of size efficient but they are more efferent in terms of weight since they are just empty sacs instead filled in organs like lungs In contrast, bats achieve the same levels of oxygen exchange as birds just by using bigger lungs.
- Bird lungs have about 2x as much exchange surface for their size as mammal lungs.
So what all of this tells us is that the efficiency you see come directly out of pneumatic bones in terms of respiration is that it allows you to reduce the needed weight and body cavity space of your respiratory system by ~25%. Everything else that is beneficial about a bird's respiratory system either does not contribute directly to respiration efficiency or it could be easily replicated just by making lungs bigger, have more surface area, or by using air sacs purely inside the body cavity.
Solution: Give them a higher Myoglobin based Oxygen Intake Mechanism
All this said, both mammals and birds tend towards hemoglobin based oxygen intake. Most animals prefer hemoglobin because it forms weaker oxygen bonds than myoglobin making it require less energy to release the oxygen when needed. However, the stronger bonding force of myoglobin means you can uptake oxygen several times more quickly and completely. In fact, some mammals have been shown to boost how much myoglobin they use for oxygen transport and absorption when injured, living at higher altitudes, or requiring a more energetic lifestyle to help compensate for higher oxygen needs; so, the mechanism required is already in place. You just need to activate it.
While hemoglobin only allows animals to absorbs ~15% of the oxygen it breaths in, mammals which use primarily myoglobin based intake (like Whales) are able to absorb up 90% of the oxygen they inhale. So, by increasing your myoglobin dependence by just a little bit, you can increase your oxygen intake by 25% to compensate for not using pneumatic bones as part of your respiratory system. In fact, if you were to switch to a purely myoglobin based exchange system, and increase the surface area of your lungs to the same density as a bird's you could achieve the same total oxygen exchange as a bird gets without even needing to make your lungs any bigger.
Other reasons to favor pneumatic bones.
The big downside, and thus the reason most organisms don't do this, is that it will take more calories to release the oxygen when needed. While inefficient calorie use is normally selected against, your creatures may be in a situation where they are genetically engineered and/or fulfill a niche where the extra calorie needs are not a major constraint.
Furthermore, most terrestrial body plans would overheat using this kind of respiratory system. Part of what a bird uses its pneumatic bones for is temperature control. Burning more oxygen means more body heat; so, birds use these bones as heat sinks to help remove this extra heat from their bodies.
So, if you are going for a body plan that prefers solid bones, you should consider that your organism may need to live in a colder environment, and/or have specialized structures that act as heat sinks like an elephant's ears or a bat's wings.