There are definitely privacy concerns.
Being signed in does ad convenience:
such as quicker results, finding open tabs on your other devices, more relevant search results, bookmarks saved across all devices etc.
The Privacy issues are frightening to say the least.
Signing in makes it easier for Google to get a complete profile of you,
The NSA can also keep track of you much easier.
Web History, cookies, you name it everything!!! is recorded on google servers.
INCOGNITO browsing is supposubly not recording your history however I believe that it does and there is probably evidence released by Snowden that proves it does.
Your GMAIL is your Google account, so in google chrome everything thing you do is being recorded on google servers with your name attached.
You are signed in!
NOTE:
Don't trust HTTPS or 256 BIT encryption anymore.
Whilst you should never sign in to sites without HTTPS, online companies, web providers, NSA etc can still have ways around them.
And if you think that there is nothing that they are interested in you, that is naive and you should really think about it because I bet with in 5 minutes of looking through your entire history there would be something incriminating.
Privacy is a human right and it is a false trade off to say that terrorism is an excuse to monitor and scan everyone's data. Dictatorships start on this very excuse, principle or what ever you wan't to call it.
The best way regarding browsers to make privacy better is to use a non proprietary browser.
This doesn't mean using "open source"!
Open Source is a falsely advertised version of freedom.
"Free Software" is the best way to go! That is why companies like Microsoft, Apple etc attack them.
The most common example of free software is "Mozilla Firefox".
Mozilla Firefox however, has non free proprietary add ons that you can install which does in deed defeat the purpose of using it as a free browser.
For software to be Free (not free beer, freedom of speech) it must contain a license like the one below.
https://www.gnu.org/licenses/gpl.html
Whilst Chrome is definitely the fastest browser known to mankind, you are sacrificing many different freedoms by using it.
Might I suggest using "tor" If you are deeply concerned about privacy.
https://www.torproject.org/
Here are some useful links:
https://www.gnu.org/philosophy/free-sw.html
I recommend installing HTTPS Everywhere
https://www.eff.org/https-everywhere
Ad block:
https://getadblock.com/
and an extension called "No Script"
https://addons.mozilla.org/en-US/firefox/addon/noscript/
Those three extensions will help improve your privacy
(my website below)
http://www.fusiontv1.site40.net/Fusiontv1/TV_Video/Entries/2013/12/16_Stop_Watching_Us_The_Video.html