In Zotero, if you drag PDFs straight into your library you still need to right-click and retrieve metadata. In general though, it's advisable to get items into your library using Zotero connectors (which work on just about any scientific journal website & repository) or using the DOI. This is because retrieving metadata (for Mendeley and Zotero alike) often relies on fairly superficial matching and won't get you all the metadata that is available through other means.
Like mark says, after you drag a PDF into Zotero, you need to right-click on it and Retrieve Metadata to attach it to a proper Zotero item for sorting, searching, citing, etc. However, the Retrieve Metadata function relies primarily on Google Scholar's full-text database, which doesn't include DOI information, so this function will rarely yield the best metadata.
In general, to import items into Zotero, you want to use the Save to Zotero button in your browser from the publisher webpage. This is both easier than first downloading the PDF and will yield the best metadata.
Although you will still get the best metadata by using the Zotero browser button at the publisher website, Zotero's PDF metadata retrieval has been much improved since January and should now get abstracts and DOIs for most articles.
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In general, to import items into Zotero, you want to use the Save to Zotero button in your browser from the publisher webpage. This is both easier than first downloading the PDF and will yield the best metadata.
See here for details https://www.zotero.org/support/getting_stuff_into_your_library