Manuscripts for scientific peer-review are delivered as PDF files. Thus it appears most natural to comment the PDF file itself instead of submitting the comments as a separate text (file). However, many submission systems do not allow to submit comments in form of an annotated PDF file.
In addition, there is no easy-to-use free/Open Source PDF editor for Linux (my platform of choice). Yes, there is PDFEdit (http://pdfedit.cz/en/index.html), but it requires substantial learning and its last release dates back to 2012. There are free online PDF editors (e.g. https://www.pdfescape.com/), but sometimes I need a local tool. Although not Open Source and not free, the tool of my choice has been PDF Studio Pro. It has all the features I need (actually much more than I need), it is truly cross-platform (Windows, Mac, Linux), easy to use and very affordable for what it offers ($129 single permanent license). Unfortunately, Quoppa software - the maker of PDF Studio - does not offer academic discounts.
PDF Studio Viewer
A while ago, the makers of PDF Studio started to offer a free version called PDF Studio Viewer. This free version has gotten more useful over time. Since last year, it also supports PDF annotation, which is very important for my work, mostly when I am peer-reviewing scientific manuscripts. For many academic users, the PDF Studio Viewer might be fully sufficient.
Libre Office and Inkscape
Of course Libre Office Draw and Inkscape can edit PDFs, but they are not specialized for annotating. If you need to do extensive annotations, the process becomes soon very painful. In addition, Inkscape editing can be destructive (i.e. when you modify text), but I have used it e.g. to fill out forms.
Apart from the lack of dedicated annotation and reviewing tools ("markups" like "replace text", "crossout text", "delete text", "insert text" or callouts), Libre Office Draw is actually a quite capable PDF editor, but fails still sometimes to correctly open very complex PDF documents (I have had problems with background images/patterns). Strangely, the only reviewing tool (comments) are not exported by default. You need to check the "Export comments" box when you export your edited PDF file as PDF (the "Save" operation creates an ODG file, which you probably don't want).
If you do not have the need to annotate, then you might get away with the Open Source tool PDFsam. Unlike PDFEdit, PDFsam is available from the Ubuntu repositories. PDFsam also has two non-free versions (PDFsam Enhanced and PDFsam Visual). PDFsam Basic (the Open Source version) is just a simple GUI on top of some command line utilities, whereas the PDF Studio Viewer offers a true visual editing experience. I canot talk about the non-free versions of PDFsam as they are not available as free trials. In fact, all of the operations of PDFsam can be achieved via the command line (see here for my blog post about how to perform common PDF editing tasks using mostly the command line tool pdftk).
Reducing file size
PDFsam Basic and PDF Studio Viewer do not offer any functionality to reduce file size. The LibreOffice Draw PDF export dialog let's you reduce image resolution and JPEG compression, which you can use to reduce file size. But at the Open Source front, the only capable tools to reduce PDF size seem to be command line tools. I use ghostscript for this task, but the command is not easy to remember:
gs -sDEVICE=pdfwrite -dCompatibilityLevel=1.4 -dPDFSETTINGS=/screen -dNOPAUSE -dQUIET -dBATCH -sOutputFile=output.pdf input.pdf
With PDF Studio Pro, you obviously do not need to remember the different keywords for the different output quality option (screen, ebook, printer, prepress) as you just choose from the drop down menu between the available options. If you are looking for an Open Source graphical wrapper for ghostscript, perhaps try workerPdf.
Signing documents and adding signatures
PDF Studio Viewer let's you sign documents if you have a DocuSign subscription. However, I am mostly concerned about being able to prove myself that I created a certain document ("self-signed signature") and not that others are able to verify my authorship. For this scenario you are out of luck with PDF Studio Viewer. And apparently, PDFSam does not offer any possibility for signing (neither self-signing nor 3rd party signing). However, Libre Office Draw allows documents signing! I have been looking for an affordable solution (not self-signed, but trusted by other PDF readers) to sign PDF documents, but there seems to be no appropriate solution if you need to sign only rarely. The basic plan by DocuSign ($10/month) appears too expensive when signing only one document per month.
If you want to stay with free or Open Source solutions, you probably need to combine several tools in order to cover all typical PDF editing tasks without pain: PDFsam Basic, LibreOffice Draw, PDF Studio Viewer and ghostscript. But if you edit PDFs often (as I do), buying PDF Studio Pro is clearly the way to go.