A couple of options work for what you're asking, however, depending on location, some may be easier to do than others. I'd successfully hidden in plain sight on many occasions, but only really have success using the "Blend In" method.
Central Clue Location:
An easy way to be able to observe all the puzzles is pick a central location for all the puzzles which has a position that would allow you to remain hidden. Those circular clothes racks at a shopping mall are good for this, as you can remain hidden and yet still able to see and hear around you. All the puzzle items would need to be in the area, but a well placed clothes rack will allow you to see/hear a great distance. Other options are available, with the key being that you need a hiding location near the final clues. *Side note: this would be better done at non-peak times, as someone hiding in a clothes rack might draw the wrong kind of attention.
Crowed Location/Blend In:
Now this method will require you practice this at times when you're not together. I'd suggest using a family member, which is something I have done in the past. Pick a busy location and time to play this game with your friend. (Think Black Friday Shopping Center) Bring a light jacket and turn it inside-out and tie it around your waist. Wear a baseball cap, but have it turned around. When the game starts, don the jacket after turning it right side in and turn your baseball cap forwards. You've successfully destroyed your friend's mental image of what you're wearing, but now you'll have to alter how you walk. Personally, I choose to take shorter more meandering strides as I'm a tall person. Hunching over can help with this. Since you're in a location with lots of distractions you'll be able to follow your friend from a slight distance. One thing to keep in mind is never make eye contact/look directly at your friend. A key to this is not to make "sudden" movements. Everything must be slow and deliberate. I've pickpocketed family members and planted notes doing this, even when they've been looking for me. I'm sure there are counter-terrorist tactic books that delve into this more.
The focal point of this method is to go with the flow of the crowd. Anything you do that is contrary to the crowd will single you out. If your friend veers off suddenly, don't immediately follow. Wait and move deliberately in his direction, but with a "target" of something different. You'd be surprised how easy it is to blend in with the "masses" and people will overlook you.
Using the above method of blending in, you can take this one step further and get other people involved to allow you to stay extremely close to your friend. Sequester random strangers to ask your friend questions. A easy example is to tell a sales associate that your friend is wanting to look at a certain item. As your friend is distracted by the sales person, you will be able to observe them closely. A dangerous/risky example would be to tell security at the mall/location you are at, that your friend has been acting strangely. Don't mention that he is your friend. Now this comes with some risks, mainly that if you involve security, you might end up in hot water yourself, but might be worth it, depending on how badly you want to win. Any outside source of distraction to your friend's "search" for you, will allow you to be close by, without being seen. I'd combine this with changing your appearance slightly for best results.
For all these to work well, your best bet is to stay out of his/her line of sight at all times and when/if you're in their line of sight, blend in with your environment.
The same can be said if in the woods. However, in nature, one big difference is that there isn't as much movement in the woods. Movement should only be done when your friend isn't looking. Beyond that, remain still and out of line of sight. Avoid bright colors as well. In the city, you need to be moving, as people in public are typically moving at relatively the same speed.