We have non-competition clauses in shareholder agreement which prevents any shareholder to own shares in a company which is a competitor. We founding partners think this is a good idea to prevent any important information flowing to competing companies but one of the shareholders who is not a founding partner wants to be able to invest in companies that may compete.
The reason is this shareholder has a portfolio of startup investments and may add more those in the future. Should we founding partners be worried if we allow that possibility? Is it typical that one investor can invest in competing start-up companies?
I have never heard of a "non-competition" agreement for shareholders. You can sign them up for NDA, but why would you limit their investment options? It is quite common that investors invest in more than one company, and checking who competes with who might be a burden not worth dealing with. If they need to drop someone, it will probably be you, not the other options, as you're the ones causing the problem.
Yes, there is precedence in law that some non-competes can be defended in shareholder agreements, however, the state chosen for incorporation and whether or not the company is considered "closely held" will largely determine the outcome if you have to defend the agreement.
If your business is found to be closely held in your state, you will find that shareholders are typically held to Fiduciary Duties, including the Duty of Loyalty not to compete. Without the non-compete agreement, your potential investor would not be able to start a competing business anyway. Officers, and directors of the business are also bound to the same duties.
Again, this depends on the state where you are incorporated. In Delaware, for instance, minority shareholders do not owe fiduciary duties to the company nor to other shareholders, even if specifically elected to be a closely held corporation, and are free to compete if they owe no other fiduciary duties through their roles within the operations of the business (director or officer).
You might consider your inclusion of a non-compete in the shareholder agreement to be a psychological tactic as much as a practical one.
In terms of is it common? It's not. Typically, you should have no concern if an investor is not part of day to day operations, nor part of the board. A shareholder in a business typically is not privileged to receive insights about the operations, IP, or protected interests of the business.