First, let's face the ugly truth: screeching "Mine!" would be a behavioral upgrade for many of our preschoolers. When we hear a young child patter over to a peer who is happily engaged in play with some fascinating object and NOT grab the goods, something already has been accomplished. If just a snatch and run occurs, we're still doing pretty well, since no toddler-on-toddler violence has been perpetrated. Likewise, if "Mine!" comes in defense of a possession, we can be glad mouths are being used for words, not biting.
Of course, it's laughable when the robber claims "Mine!" about the goods in question, but it represents a crucial first step toward articulate self-expression and negotiation. We want to coax our kids out of their adversarial stance into something a little more humane, urbane, and, well, sane. "Mine!" is the simplest articulation of wanting, and a critical step in the progression toward "Me too!," "Can I have a turn?," or, best yet, "Let me know when you're done with that, okay?" We all know what it's like to want something and it's our job as grown-ups to help our children use language to get it. They must view language as the means to express what they want and negotiate procuring it in light of what others want.
Encourage your toddler to speak as the means to resolve conflicts. Give them opportunities to express what they are feeling. Don't be tempted to steer children off to some other distraction without getting to say what they have to say. But also teach them that just because they want it doesn't mean they are going to get it, or at least right now. Above all, they need to see you model the right way to conduct themselves in interpersonal interactions. Otherwise, they might model themselves after what they see on Jerry Springer or in Congress.