9:26 p.m. on December 27, 2012 (EST)
iClimb,
That's a question you can only answer for yourself. The way to answer it is the same advice given many times under the Beginners and Backcountry forums - start with short, simple, easy routes and work your way up. Do some of the same practice scenarios you got in the training classes with the guide/training organizations.
Obviously, in most cases, you don't want to go solo in the big mountains and remote places. So you do want your companions to have similar or more extensive experience, and you want to have been on easier trips with them - it is really interesting to find out how different some people are after a couple weeks into a challenging expedition. I have had the experience a couple times of having someone in the party who we were all asking "whose friend is he?" or "who invited him along?" (one of those was the nominal expedition leader). I had a tentmate say, about the 3rd day sitting out a storm in the tent, "This is like being in prison, except you can't escape!" By the end of the trip, the rest of us were about ready to toss him into the nearest crevasse.
The NH Whites and the North Cascades are quite different in their challenges. The Whites have lots of steep ice and unexpected storms. The North Cascades have lots of heavily crevassed glaciers with lots of hidden crevasses covered by thin snow bridges and unexpected storms.
One suggestion is to head out west and spend a month doing climbs in the Cascades. Shasta is a good starting point with Avalanche Gulch a "beginner" route, Whitney Glacier a bit more challenging, and the routes on the north side going all the way up to large crevasses and overhanging ice. Getting into Oregon, South and Middle Sister are basically hikes, Hood is more challenging with some glacier routes with active crevasses (people have gotten thoroughly lost on the "standard", "dog route" when coming down after dark or in whiteouts, plus has serious unexpected storms). I do not recommend doing North Sister, though - lots of rotten, loose rock. In Washington, Adams is a long, though basically hiking route by the "standard" route (people have gotten thoroughly lost on that route in whiteouts and staying out too long after dark). Baker is a good intermediate mountain by several routes from the north, with lots of crevasses, and with ice climbing by the north ridge. Rainier can be done from the Paradise side (I don't recommend that route, since that's the one "everyone" takes, especially the RMI huge guided parties - don't get stuck behind them!). Better is the approach from Sunrise, with a night in Glacier Basin to acclimatize and do some refresher practice with self-arrest, then a night at Camp Schurman, where you can do some "for real" crevasse rescue refresher practice, then off up the mountain at midnight (the guided parties mostly stop in the crater, so you are likely to get the summit to yourself).
If you work your way up, step at a time, you will be able to readily judge for yourself your readiness.
One thing I might note, since you mention the potential of going abroad - the logistics of doing really big mountains is such that sometimes it is advisable to use a guide service. For example, going to South America, you will need to hire arrieros (muleteers) to haul the multi-day supplies for the approaches. If your Spanish is fluent, you can deal with the arrieros and porters (although some tend to speak in their local languages - Quechua and Aymara in the Peruvian Andes, for example). In some areas, you may take a taxi to the higher villages or even partway up the valleys, then pick up porters. In Patagonia, the climbs of interest are mostly serious rock climbs, which you can approach and do on your own if you can climb 5.9 and above in storms, but the logistical challenge is getting from the closest airport to Paine and Fitzroy (bus, then taxi, though they speak fairly standard Spanish - except the Chileans tend to drop final syllables and sometimes a few words, since "everybody" knows what they are talking about).
In Alaska and the Yukon, you can, with experience, get a bush pilot to drop you and your supplies off, then haul your loads (multi-carries, since you will be carrying a full month's supplies) up to your high camp (you only have to fight the B29-sized mosquitoes up to 4000 ft or so, which means no real problems for the Alaska Range, Wrangell, and so on since you will be landed on a glacier - if you think I am joking about the mosquitoes, just ask anyone who has been in the Arctic).
All of this has a BIG caveat - you need to be ready to deal with being completely on your own. While you can get rescued in a couple of days if you get caught in a storm on Rainier or Hood, it may take a week or so for problems on Denali or in the Sheldon or Ruth Amphitheaters, and even longer if you are in the Brooks Range or Wrangell-St Elias area. I have had to wait close to a week to get in and similarly to get out up there, due to bad weather. Really remote places like Antarctica, much of the Himalaya, and even much of the Andes don't have even that much plane access.
There are some big mountains in certain 3rd World countries where you are required to hire local guides and porters, although for many of those, the "guides" are only to get you to the base of the climbing area and can not guide you on the climbs - Mt. Kenya, for example.
On the other hand, some of the most rewarding and fun time I have spent in the mountains has been when off with a few friends, knowing that we wouldn't be retrieved for 3 or 4 weeks, and sitting out a multi-day storm.