I have just recently filed a DMCA notice on this blog for copyright infringement of my blog. So far nothing has been done except receiving an automatic reply from Google. When will the other blog be taken down?This blog owner does not understand that not all legal actions can be executed immediately - and some, never.
All legal complaints, registered by you or someone else, may not always produce the desired results.
All complaints require some time for action, and some require more time.
Some DMCA violations are processed in days, others take weeks. Every blog owner, on either side of the complaint, has legal rights - so the rights of the other blog owner have to be considered.
Some blog owners attract multiple complaints, and Google may have escalated the case to another team. Larger cases may justify longer preparation. Larger blogs may take longer for analysis.
Some complaints may lack necessary detail.
It's also possible that the complaint lacked clarity. Some complaints look obvious to the offended party (ie, you) - but are not so obvious, to Google Legal experts.
Some complaints require subject familiarity with the blog being evaluated.
Complaints about a blog in a non familiar culture or subject - or published in an unfamiliar language - may require services of an expert in the culture, language, or subject. Not all experts, in all cultures, languages, or subjects are available, at a moment's notice.
Not all complaints are genuine.
Finally, some complaints are just bogus. Some people use DMCA complaints as anonymous attacks, against innocent parties.
Each complaint has to be carefully evaluated, it can't be taken at face value, blindly. Evaluation will take time.