I am sure that Spotted by Locals is mostly read by ecologically, sustainable, globally yet locally minded people. People who like things, but not their impact on our one and only Earth. If my guess is correct, then for souvenirs you should head to Uuskasutuskeskus – re-use store that collects stuff that people just do not need any more and give it up for free or sell it at very low prices.
There are several stores in Tallinn, but the most central and biggest is at Tatari 64. So, what you can expect – lots and lots of old dishes and vodka shot glasses (that along with some local quality vodkas such as Moe would make a perfectly reasonable gift), books (in a bizarre Estonian language that actually could also come in handy after previously recommended items have been consumed), clothes and furniture (in case you did not arrive by Ryanair), truly local people and perhaps lost time.
To cut the spending on the latter, take tram 3 or 4, heading from the centre towards Tondi. Exit at the stop Vineeri and it is just 100 steps aways.