For plasmid vector, linearization is the indication of complete digestion, but how to check whether the insert which is a linear PCR amplified fragment is completely digested?
You could use a small amount of your digested PCR product as template in a PCR with the same primers. A successful digest should have destroyed all the template molecules. Therefore, your PCR should not lead to product formation (do not forget controls!). Your digest was not complete if you get products in this PCR. Unfortunately, it is not a perfect assay, but I hope it helps you.
If I understood well, then the primers can still bind to the digested insert. The insert has restriction sites (a part of the primer) at both ends. So the construct is RE1-insert-RE2, where RE 1 and 2 are two different restriction enzyme sites flanking the insert. When I amplify this construct and digest it with RE1 and RE2 and use the same primers (used to insert the RE1 and 2 sites) for PCR, I think I'll still get the amplicon. Shouldn't I?
Yes, you are right. I thought about something different. However, you could still try a PCR to check the digest. It could be possible to distinguish between digested and complete PCR product if you increase the annealing temperature. At some point a primer will only bind to the complete PCR product. But this is only possible if there are some additional nucleotides next to the restriction site outside of your product. Nevertheless, this is really complicated and probably not helpful.
You should just incubate over night using high fidelity enzymes. Unfortunately, the successful construction of your desired plasmid is the only way to show the complete or at least sufficient digestion.