There is no west Bengal dialect per say, Bangla is a very loose language bound together by a set of rules as such there are alot of dialects and sub dialects.
West Bengal has many dialects, I don't know much about them but I can surmise they're as diverse as those in Bangladesh. Perhaps
@Joe Shearer would know better on this.
West Bengal Bengali uses Sanskrit words alot more in Bangladesh, much of these words are considered 'archaic'.
Bangladeshi Bangla is also very diverse with many dialects, two of these dialects are almost unintelligible, namely Sylheti and Chattgaiya or Chittagonian.
Chattgaiya is a language that can only be spoken not written, the perso-arabic based script is now lost. I speak Chittagonian. It is spoken in the Chittagong region and by the tribals aswell.
Sylheti is spoken in sylheti is one of the more well known Bangla dialects because it has been perpetuated abroad by sylheti immigrants in the United Kingdom and United States.
People from other regions of Bangladesh will have a hard time understanding either of the dialects but you can still get the gist of what a person is saying if you listen closely.
Other than these two major dialects, the others can be grouped together and they're for the most part mutually intelligible apart from a few region specific words and sayings.
There is also the literary Bangla ( Bangladesh) that is the 'pure' or 'shuddo' form of Bangla, everyone in Bangladesh is capable of speaking it to variable degrees. This what you hear the anchors on TV speak, even then they're not speaking it well.
Coming to whether WB bangla and Bangladesh Bangla are mutually intelligible., For the most part, yes they're mutually intelligible, provided we don't take dialects like sylheti or Chittagonian into account.
In Bangladesh, we think people of WB have an accent, don't know if that's true or if we just perceive it that way. I will say this WB people have a 'Hindi' esque flow to their speech.
People of WB think we in Bangladesh have an accent.
I would say that, people may need a minute or two to calibrate themselves to understand people from the other side of the border, if they have never heard people from the other side talking before.
Anyway, these are my personal observations, Bangla is at the core Bengali culture and is a very sensitive issue and it's easy to offend people, if you unintentionally claim to have the better, clearer diaclect.