This is possible to a limited degree assuming you control the content that they are hosting.
The good news is that with the ObjectForScripting property you can pass data back and forth from javascript to .net and then again so that one web browser can react to the other.
The other bit of good news is that they are going to use the same WinInet cache. So if you create a cookie for the site, both browsers could potentially read it if it was a session cookie or such.
Although I would have to know a bit more about what you had in mind to give you a better and more detailed answer.
Here is some sample code nabbed from the MSDN on objectforscipting that shows how to interact with a single web browser control
Imports System
Imports System.Windows.Forms
Imports System.Security.Permissions
<PermissionSet(SecurityAction.Demand, Name:="FullTrust")>
<System.Runtime.InteropServices.ComVisibleAttribute(True)>
Public Class Form1
Inherits Form
Private webBrowser1 As New WebBrowser()
Private WithEvents button1 As New Button()
<STAThread()>
Public Shared Sub Main()
Application.EnableVisualStyles()
Application.Run(New Form1())
End Sub
Public Sub New()
button1.Text = "call script code from client code"
button1.Dock = DockStyle.Top
webBrowser1.Dock = DockStyle.Fill
Controls.Add(webBrowser1)
Controls.Add(button1)
End Sub
Private Sub Form1_Load(ByVal sender As Object, ByVal e As EventArgs) _
Handles Me.Load
webBrowser1.AllowWebBrowserDrop = False
webBrowser1.IsWebBrowserContextMenuEnabled = False
webBrowser1.WebBrowserShortcutsEnabled = False
webBrowser1.ObjectForScripting = Me
' Uncomment the following line when you are finished debugging.
'webBrowser1.ScriptErrorsSuppressed = True
webBrowser1.DocumentText =
"<html><head><script>" &
"function test(message) { alert(message); }" &
"</script></head><body><button " &
"onclick=""window.external.Test('called from script code')"" > " &
"call client code from script code</button>" &
"</body></html>"
End Sub
Public Sub Test(ByVal message As String)
MessageBox.Show(message, "client code")
End Sub
Private Sub button1_Click(ByVal sender As Object, ByVal e As EventArgs) _
Handles button1.Click
webBrowser1.Document.InvokeScript("test",
New String() {"called from client code"})
End Sub
End Class