In 2007, we were hiking back from Grinnel Glacier in the late afternoon.
Towards the bottom of the glacier trail, where it meets up with the trail along the north shore of Lake Josephine, we were warned by a ranger that there had been a grizzly bear sitting recently along the shore of the lake.
We took precautions by making more noise and being vigilant
We'd almost hiked passed the lake without seeing anything when we came upon two other hikers stopped on the trail ahead of us, looking intently down towards the water.
We walked up to them, and they said they'd seen a young grizzly go into the water very near the trail. We decided to wait it out for a few minutes before proceeding. After about 5 minutes of nothing, we made a couple of calls of "hey bear!" (to warn it of our presence) and moved on.
Where we stopped, there's a small pond completely separated from Lake Josephine that's about 40 yards long. As we got past it, I looked back over my shoulder and saw it - a brown grizzly in the water, looking back at us, maybe 40 yards away.
It didn't make any aggressive movement, though it did swim slowly towards us along the edge of the pond. We were all slightly nervous, but since we knew the grizzly bear was aware of us (and probably had been well before we knew it was there) and hadn't behaved aggressively, and looked completely calm with no signs of agitation, it was probably going to leave us alone.
I had my camera ready as soon as we'd heard from the two hikers there was a grizzly bear in the area, so we paused for a few seconds, and snapped several pictures.
When he left the water and started walking towards the trail on a path that would take him maybe 20 feet from us, we figured it was well past time to go, and continued down the trail.
100 yards or so up the trail we met several groups of hikers heading the opposite direction (I think the boat had just unloaded some hikers). We warned them there was a grizzly bear near the trail, and to be cautious.
The amusing thing was some of the reactions...
There was one couple, a 20 something guy and girl... the girl kind of freaked out and said something to the effect of "Let's get out of here!". The guy seemed very excited though, and started tugging her to head down the trail towards the bear. Not sure how that got resolved.
Overall, it was a great experience. The grizzly bear seemed completely calm and aware of us, and we got some nice pictures and a story too!
In retrospect we were probably well within the bear's "comfort zone" and probably should have been a little more concerned (ie, I should have had the safety off of my bear spray as soon as we knew there was a bear close by).
I think it helped that we were in a group (5 people including the other two hikers when we saw it), and we'd been making noise to alert it to our presence, so we didn't surprise it.