According to Google Webmaster Tools, I have a number of duplicate title tags. This happens because my products all have two URLs:
/products/product is redirecting to
/products/product-category/product, so should I worry about all the duplicate title tags? Does it matter what content is on a page that is redirected?
Its won't be any issue, but I believe instead of using 301 redirection you need to use canonical tag to avoid such duplicity issues. 301 redirection work best at domain level
Google Webmasters provides the details of the Duplicated titles present for your site.We need to download those information.Then We need to use rel="canonical" in the suitable https://googlewebmastercentral.blogspot.in/2013/04/5-common-mistakes-with-relcanonical.html way to solve this issue.
Else the Google treats it duplicate content https://moz.com/learn/seo/duplicate-content is explained clearly here with pictures.
In E-Commerce site when similar products are repeated this issue occurs.At that time rel="canonical" is used in the main page. So that Google robots treats it as main page and other similar pages are following it.http://www.goinflow.com/duplicate-content-ecommerce-seo/ . This solves the Duplicate Title tag issue & Duplicate content issue also.For dynamic content this would be https://stackoverflow.com/questions/16325842/how-to-prevent-duplicate-title-tags-on-dynamic-content helpful.
And also minimize Redirection as far as possible because it slows down site speed.(https://varvy.com/pagespeed/minimize-redirects.html). Because Page Speed is one of important ranking factor.(https://moz.com/blog/how-website-speed-actually-impacts-search-ranking) . So we need to think of Page Speed as well.