Let's start with the problem: the "Send an email" button sends an email to the recipient's Google email address. No notification shows up in Plus, which is unfair to people who don't check their email address regularly. Those people should, however, and who's to say Google can't tell them "tough luck"?
Well, I don't think Google can tell them "tough luck", because Google has to beat Facebook's network effect, and telling people "tough luck" isn't very welcoming. There are a few ways to make the emailing component of Plus more welcoming:
1) Google could imitate Facebook's messaging system. Not Facebook's current messaging system, with its lack of subject lines and strangely arranged inbox, but their old messaging system. This was my original proposal, when I was talking about the issue with my brother. To Google employees, this idea is likely to be radical, if not stupid. Thus, I defended my position in a Google+ post, excerpted here:
Next, I want to talk about Google's viewing of Google+ as another "part" of Google. I think integration is good, as long as it is useful and doesn't confuse users. I think Google+ should have an internal messaging system, just like Facebook does. Instead, they've decided to connect with Gmail. What if you have a Yahoo email? And also, an internal messaging system would be better because it's in line with people's concept of what a social network is. Obviously, a social network is really a suite of several products, e.g. Facebook has a messaging system, an instant messenger, a content stream, etc. But users of Facebook have become accustomed to viewing that as one unit. Just because in Google+ there's the toolbar up top that has a Gmail link doesn't mean users don't feel like they're going outside of Google+ (which they view as the product they're in, not "Google") when they click it. Messages from within the social network should have notifications within the social network… I am opposed to the simple fact that the wording is "Email Samuel Freilich" not "Message Samuel Freilich." Even if it did the exact same thing. I think Facebook's core features should be mimicked pretty closely, because that will make it easy for people. To reiterate, I'm not advocating for having a Yahoo email address, but if you want to get people on your social network… you need to make some compromises.
2) Google could allow users to hook their "Send an email" button to other email services, not just Gmail. This is not really a solution, because it still doesn't address people not checking their email. Before I continue, I just want to point out that I am not one of those people: I check my email every day; however, I am affected by those people: if I want to communicate with a friend, I will often not want to send them an email, because I don't know how often they check their email. I can better get a sense of or guess how often they check their Facebook, so I choose messaging them on Facebook over emailing them. Google+ needs to create a smooth transition for people migrating from Facebook.
3) Google could put a new email counter to the right of the Gmail link on the top toolbar. That way, people who use Yahoo email would be forced to click on the Gmail link to get to their emails sent to them through Google Plus. And they might even fall in love with Gmail while they're at it. That's a win for Google and it's more of a win than not for the Yahoo emailer, because at least he isn't receiving emails he doesn't know exist. The downside is this might be kind of ugly. There would be a Gmail notification counter on the left and a Google+ notification counter on the right. And power emailers, who use all kinds of filters and acrobatics in their inboxes, might get annoyed by the counter. It might be a lucrative choice though.
4) Google could notify the recipient via the Google+ notification counter. Just those emails which were sent via Plus. When you clicked on the notification, you would either be taken to Gmail itself, or the data would be pulled out of Gmail, and you would be able to see your email that way. The latter would be more comfortable and homey, and if I were building it, I'd make sure there was the ability to label, delete, and otherwise handle the email right from Plus. Which is very similar to my original suggestion.
5) What Google actually did: two buttons on the profile page, "Send a message" and "Send an email". But they didn't create a messaging system for the "Send a message" button: instead, clicking "Send a message" opens a post/status prompt and sets the privacy to [recipient's name]. Below the "Share" button, Google tells you, "Add someone's name (instead of a circle name) to post a message just to them." When you post a status with the privacy set to just one person, it gives that person a notification,* although I wish the prompt would spell that out, because it's not clear. I'm not sure there's room to tell you though, from a design perspective. The other issue I have with the new "Send a message" button is the "message" appears in the stream not an inbox. Somebody could easily be looking over your shoulder when your scrolling through your Google+ stream. You could be scrolling through your stream in order to share a link with a friend who's in the room.
In conclusion, the new "Send a message" button is an improvement, but it's not good enough. You've got to have an actual messaging system. It can have Gmail integration, but I don't know that it's a necessity.
* Actually, notifications are generated for all individual names typed into the visibility box. You can type in multiple names; the recipients will still be notified.