"Some" just means a sub-set with at least one member.
If "all dogs" are thought of visually (like a Venn diagram) as a circle, then "some dogs that are carnivorous" would be a smaller circle inside the "all dogs" circle. Likewise, "some dogs that are vegetarian" would also be a smaller circle inside the "all dogs" circle.
Do the two smaller circles inside the "all dogs" circle overlap?
Well, if we take carnivorous to mean "eats meat" and vegetarian to mean "eats vegetables" in an exclusive sense of "eats only meat" and "eats only vegetables" then they do not overlap. If we take carnivorous and vegetarian to describe the predominant diet, then they could overlap at different times of at least one dogs life, but this is stretching things a bit far to make a contradiction plausible. Also, where they might overlap, would we not say the dog is omnivorous?
As for whether or not each premise is true, there is significant evidence supporting the claim that some dogs are indeed carnivorous. While dogs are generally considered to be omnivorous, I don't think there are many considered vegetarian, tho it is certainly arguable that there is at least one instance of a vegetarian dog somewhere.
So, if we take premise one to be true and premise two to be true, are we correct in deducing that there is a dog that is both vegetarian and carnivorous? In the exclusive sense of "only eats meat" or "only eats vegetables" this is obviously a contradiction, so no, the two smaller circles do not overlap. In the loose sense of "predominant diet" then the truth value is still relative to the time of evaluation and predominance implies that one diet or the other is the main source of nutrition, so, again, no: there are no dogs that are both carnivorous and vegetarian.
Of note, exclusively vegetarian diets are not at all a good idea for your doggy. Please talk to a licensed veterinarian for nutritional guidelines and breed risk assessment if you are feeding your dog like so.
As for logic, if you dig this kind of stuff, check out Logic-Cola by Harry Gensler - it is a fun companion to his book, Introduction to Logic.
Let's look at your initial premises:
Premise 1) Some X are A (i.e. at least one X is A)
Premise 2) Some X are B (i.e. at least one X is B)
Here I think you might want an additional step, stating the relationship of A and B.
Premise 3) All A are not B (i.e. No A is B)
Conclusion) Some X are A and B (i.e. at least one X is A and B)
is false because it contradicts the third premise.