How to create a collection of anonymous sequences for teaching and testing?
There are a couple key words in this question - anonymous and teaching. Yes, NCBI is a source for sequence data, but it is not anonymous (it is annotated, which means a student could also find it and copy/paste that annotation without doing the actual analysis). Notice, I am not assuming that the request is for human data. Now, if anonymous human data are needed, most available sequence data are anonymous but there remains that annotation issue: If it is annotated already, what will they learn?

A good alternate source for some human genome data would be from Complete Genomics. They have released anonymized (de-identified) data for at least 69 subjects. The question asks about 1000 sequences, but how large? This is an important consideration. Other details are also lacking in the question.

Another source may be the 1000 Genomes data, also human. If you're interested in plants, there are sequence data out there from ~98 different Arabidopsis thaliana accessions/cultivars/strains.