Just like with correction of facial folds, temporary fillers used for reshaping the nose gradually dissolve and the correction slowly fades away. The timing of re-injection can be either after all the product has dissipated or before it's gone away completely. Either approach is reasonable.
The only situation where it is critical to let all of a filler resorb is if a patient is finally opting for a surgical rhinoplasty. This ensures that the surgical plan is tailored to the actual underlying anatomy and isn't going to change substantially afterward. With Radiesse, for example, I recommend my patients have waited at least 2 years from a Radiesse injection before considering surgical rhinoplasty.
The only permanent filler approved by the FDA for cosmetic use is Artefill. Its approval is for correction of nasolabial folds (a.k.a., smile lines or "parentheses"), but that makes it quite legal to use "off-label" for nasal augmentation.
As for the statement that permanent injectable fillers thin "your nasal skin, cartilage, and bones"...well, there's simply no evidence for that whatsoever.
The biggest risk with Artefill, for example, is that of granuloma formation in about 1 in 800 patients (0.125%). Granulomas are hard to treat and may require surgical intervention in severe cases. Of course, contrast that with the fact that 1 in 10 to 1 in 20 surgical rhinoplasty patients (5%-10%) seek revision (i.e., surgical intervention).
All the best,