The first thing to think about when you think Privacy is the operating system (OS) you use. If you're using an popular OS, that leaves you open for being a target by the hacker community. If you're using an OS which doesn't care about your privacy, such as Windows 10/8/7, you risk your information being seen by administrators of the servers or by hackers who break in.
For every operating system, if it's not the one you use you will have a learning curve to learn how to use it and find new Applications to replace other ones you used.
Here are some of my choices for an Operating System.
Apple does a good job at privacy and security by providing privacy tools, asking up front if you want to enable something that may infringe on your privacy, providing full drive encryption, preventing random applications you download off the internet from running unless Apple itself is aware (white listing), and providing built in anti malware which can be updated on the spot by Apple to remove newly found malware or prevent them from being installed.
There are some disadvantages to using OS X though. You are pretty much stuck with buying or having a Mac to run OS X. You can get around this with what is called hackintoshing, but you have to understand UNIX pretty well to do so. Being one of the top operating systems, there are some hackers that are looking into vulnerabilities which Apple isn't exactly quick to releasing patches. Apple usually waits until they have a lot of things to fix before they release a patch for the OS.
Some advantages are because it's a popular OS, lots of programs are being developed for it and you can even run some Linux applications on it using Xorg.
This OS has no issues with privacy as far as I am aware and in-fact has a privacy enhancing mode which makes things you do, such as browsing history, temporary. It acts as an OS and nothing more.
One thing Elementary OS does well is look pretty. It is based on Ubuntu which is based on Debian so you get all the benefits of both in one. You can configure it so that the Dock is auto hidden or auto shown so it doesn't take space on screen giving you a lot of space for windows.
Because Linux isn't popular, you will not have as many programs as are available for OS X and Windows, but you can use Wine to run some Windows programs or even run Windows in a virtual machine to run all Windows programs. Elementary OS comes with the Ubuntu Software Center which is a directory of programs you can install which makes finding alternatives easy.
You get amazing security using this because it is Linux and any security vulnerabilities are patched quickly by Debian.
Same as Elementary OS but with a more Windows like UI. The different versions are just different UIs available, I encourage you to play with the different versions before using. If you would prefer not to be based on Ubuntu, but based on Debian itself you can download the Debian Edition http://www.linuxmint.com/download_lmde.php
Ubuntu does have some Privacy issues out of the box, but you can easily disable that issue via the privacy settings. By default Ubuntu includes web results when searching in the Unity search box, this both slows down results and is a privacy issue because everything you type in the search box goes to Canonical (the parent company).
Ubuntu sort of mixes Mac and Windows into it's own thing as far as UI goes. Everything else is the same as Linux Mint and Elementary OS.
The great thing about Ubuntu is there is lots of support for it. It's hard to find software that will not work on it. If you have trouble, there is a great community ready to help.
If you want Ubuntu without the UI provided and without the privacy issue, Ubuntu Gnome (https://wiki.ubuntu.com/UbuntuGNOME/) exists which uses the Gnome UI. Xubuntu (http://xubuntu.org/), and Kubuntu (http://kubuntu.org/) also exists which uses the XFCE UI.
This OS is more for the geeky community. It's more bare bones and doesn't come with a software center making it harder to find software for the normal user of a computer. It has great privacy and provides multiple UIs, I recommend you try the different UIs in a virtual machine before using. I personally like Gnome and the standard Debian UI which is chosen by default.
For the true geek. You have to know Linux to install the OS. They have tutorials that teaches you what you need to know to install, but you will likely get lost if you don't know much terminal. You start off in a terminal, you have to format and install onto a hard drive manually, you have to configure the internet manually, you have to install a UI manually, it's really is for the true geek. You get the best privacy and security with this OS though because they have packages always up to date which means you get security updates really quick. Some things requires you to use the user repository which is hard to use unless you have yaourt installed.
I have not tried NetBSD, PC-BSD or OpenBSD, but I'm assuming it's similar to FreeBSD. These are true UNIX solutions which means many of the programs that works on Linux will also work on these systems with minor or no changes to the code. The only issue I have with BSD is I do not know how well support is as it's even less popular than Linux. Mac OS X has some things based on FreeBSD like the user space and Juniper (a commercial router company), and Netflix uses FreeBSD as their core OS.
From what I've seen, installation is similar to that of Arch Linux, so it's not made for the common user who doesn't know much about terminal.
For the best privacy, I recommend Arch, BSD, Debian, Linux Mint, or Elementary OS.
You can test any of the Linux operating systems I have specified with Virtual Box (https://www.virtualbox.org/) which is an open source virtual machine environment for almost every OS.
Before installing, make sure to backup any important files as you may accidentally or intentionally delete them.
When you want to install to a system, simply burn the ISO to a disk or use YUMI (http://www.pendrivelinux.com/yumi-multiboot-usb-creator/) to put onto a USB drive and boot off the installation media.
Installation steps are different if you want to dual boot with your old OS or if you want to single boot into the newer one. Single boot is always the easiest method.