For the OP some suggestions:
1. Two books have been HUGE for helping me get our DS (now 3.4) talking more. "Play To Talk" and "It Takes Two To Talk" are great resources that have specific details about games and interactive styles. The things they suggest have made a big difference for DS and have encouraged a burst of language. Plus they are fun :)
2. Pictures are great. For our DS we have made a series of small notebooks with 4 pictures per page. So, for example, we have a food book and I have taken pictures of the actual foods he eats. So now, when he wants something to eat he can go get the food book and then point to what he wants (though to be fair he basically points to ice cream and only grudgingly points to something on the "lunch" pages when I refuse to give him what he really wants.) I've done this with my iPhone, I just take a photo with the phone then it gets automatically sent to my cloud and i get it on the computer. I use iPhoto and make a 4 per-page image, label then with the right word, print and use a quick laminating sheet to laminate it. Hole punch and pop it into the notebook. We've also got a schedule book with activities (park, pool, doctors, etc) which I use to show him where we are going or for him to pick where he wants to go when that makes sense. We also have a book/video one that allows him to pick what book he wants me to read him. These have helped his vocabulary grow leaps and bounds because I always say the word for what he points to. So he sees the picture, hears the word and sees it written. I think having photos of the real item we see in our daily lives has been so much better than some abstract line drawing of things.
3. Recasting - this is a technique where you basically follow the child's lead to help them expand their expressive language. So for example, if you DD says "car!" while you are playing, you would repeat "car" then say "car go!". Then wait and see what she says next. The general rule is to repeat then add one more word than they say, then give them a chance to respond with something. So when she starts using 2 words you would repeat those 2 words then respond with 3, etc. It can also be used when she is mispronouncing something. Rather than correct her, just recast the word correctly. So if she said "caaaa" for car, then you would just say "car! car go!" so she hears it said properly. Its simple though for me the hard part has always been waiting with an expectant look on my face. I tend to "over" talk and not really wait in silence for a little while giving DS the chance to verbalize something to me.
Regarding the testing issue. I think this sooooo complex. Our son is language delayed and we got very worried when he was about 18 months and still wasn't talking. Between 18 months and now we have been on an emotional roller coaster trying to figure out what is causing his delays. That there are a large number of children who simply talk late is not something any specialist we saw told us. That there are a million reasons for someone to talk late is also not something we were told. Instead, literally every specialist we saw told us DS is autistic we went a little crazy with a special ed preschool, therapies, etc. Of course we were trying to figure out the root cause of his inability to talk. We now know he has a fairly severe language disorder but has no other delays and is not on the spectrum. Looking back I have learned a number of things. 1. Before a child is 3 or 4 it is basically impossible to really know what is going on. There can be clues, but when kids are so neurologically immature there are many things going on and even the world's expert can't really tell you the root cause. 2. Early intervention can make a huge difference for children. BUT for children with moderate/mild delays, parents can in be just as good if not better therapists for their children with the right training and approach. Working with a professional might make things more along more quickly, but children without a true disorder (kids who are really just "delayed") with catch up on their own without intervention from a specialist.
I often wonder if the round after round of evaluations, therapies, and my own incredibly high stress level really were the best thing for DS. Looking back, I suspect that DS would be just as far along if not father if I had calmed down, did some research, learned how to work with his delay, and just spent all the time he was getting evaluated and therapies playing with him instead. There is very good data to support the notion that parents can be just as good if not better therapists for their language delayed children (see Hanen training info for some of this data).
Of course, it can be notoriously difficult to distinguish between a delay and a disorder - though one very good indicator of a simple delay is a parent or close relative who was also a "late talker". So I guess I would personally have my child checked out if they were having expressive delays at 3. In the absence of a receptive language delay and no other developmental problems, I would suspect verbal apraxia. But there is a reason that children aren't even evaluated for developmental delays until 3 or 4 in many parts of the world, and the push for early intervention is a main reason why childhood developmental disorders are being diagnosed at much higher rates. It used to be that children weren't out in the world until 5 or 6 and by then the vast majority of kids with delays (and NOT disorders) had just caught up. If you feel confident that you child is just behind, then letting them grow up a bit before throwing them into the world of therapy and labels is your choice and I'm not willing to judge someone for listening to their gut about their own child. Its not the road I would take, but I trust parents to know their children!