Init commit: worked 8.0 version, which will be changed to 9.0.

Changed __openerp__.py and README.rst only.

res_partner_mails_count/README.rst

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+Smart buttons for mails count
+=============================
+
+This module adds Smart buttons with "Mails from" and "Mails to" count of mails in the partner form.
+
+Usage
+-----
+
+You can see Smart buttons "Mails from" and "Mails to" in the contact form in the Messaging/Contacts menu. If you click on these buttons, you can see list of corresponded mails. Click on the "Send a message" link to send mail to the partner.
+
+Further information
+-------------------
+
+HTML Description: https://apps.odoo.com/apps/modules/9.0/res_partner_mails_count/
+
+Tested on Odoo 9.0

res_partner_mails_count/__init__.py

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+# -*- coding: utf-8 -*-
+import models

res_partner_mails_count/__openerp__.py

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+# -*- coding: utf-8 -*-
+{
+    "name": "Smart buttons for mails count",
+
+    "summary": """
+        This module adds Smart buttons with "Mails from" and "Mails to" count of mails in the partner form.
+    """,
+
+    "description": """
+        You can see Smart buttons "Mails from" and "Mails to" in the contact
+        form in the Messaging/Contacts menu. If you click on these buttons,
+        you can see list of corresponded mails. Click on the "Send a message"
+        link to send mail to the partner.
+    """,
+
+    "author": "IT-Projects LLC, Pavel Romanchenko",
+    "website": "http://www.it-projects.info",
+
+    # Categories can be used to filter modules in modules listing
+    # Check https://github.com/odoo/odoo/blob/master/openerp/addons/base/module/module_data.xml
+    # for the full list
+    "category": "Uncategorized",
+    "version": "1.0.0",
+
+    # any module necessary for this one to work correctly
+    "depends": ["base", "mail"],
+
+    # always loaded
+    "data": [
+        "views/res_partner_mails_count.xml",
+        "templates.xml",
+    ],
+}

res_partner_mails_count/models.py

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+# -*- coding: utf-8 -*-
+
+from openerp import models, fields, api
+
+
+class res_partner(models.Model):
+    _inherit = 'res.partner'
+    mails_to = fields.Integer(compute="_mails_to")
+    mails_from = fields.Integer(compute="_mails_from")
+
+    @api.one
+    def _mails_to(self):
+        for r in self:
+            r.mails_to = self.env['mail.message'].sudo().search_count([('notified_partner_ids', 'in', r.id)])
+
+    @api.one
+    def _mails_from(self):
+        for r in self:
+            r.mails_from = self.env['mail.message'].sudo().search_count([('author_id', '=', r.id)])

res_partner_mails_count/static/src/js/main.js

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+openerp.res_partner_mails_count = function(instance){
+	instance.mail.Wall.include({
+		init: function(){
+			this._super.apply(this, arguments);
+			if(this.context.ignore_search_model){
+				delete this.defaults.model;
+			}
+		}
+	});
+};

res_partner_mails_count/templates.xml

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+<?xml version="1.0"?>
+<openerp>
+    <data>
+        <template id="res_partner_mails_count_assets_backend"
+                  name="res_partner_mails_count_assets_backend"
+                  inherit_id="web.assets_backend">
+            <xpath expr="." position="inside">
+                <script src="/res_partner_mails_count/static/src/js/main.js"
+                        type="text/javascript"></script>
+            </xpath>
+        </template>
+
+        <record id="search_notified_partner_ids" model="ir.ui.view">
+            <field name="name">mail.message.search.notified_partner</field>
+            <field name="model">mail.message</field>
+            <field name="priority">50</field>
+            <field name="inherit_id" ref="mail.view_message_search"/>
+            <field name="arch" type="xml">
+                <search string="Messages Search">
+                    <field name="notified_partner_ids"/>
+                </search>
+            </field>
+        </record>
+
+        <record id="view_res_partner_mails_count_info_form" model="ir.ui.view">
+            <field name="name">res.partner.mails.count</field>
+            <field name="model">res.partner</field>
+            <field name="inherit_id" ref="base.view_partner_form"/>
+            <field name="priority" eval="50"/>
+            <field name="arch" type="xml">
+                <xpath expr="//div[@name='buttons']" position="inside">
+                    <button class="oe_inline oe_stat_button" type="action"
+                        name="%(action_mails)d"
+                        context="{'search_default_notified_partner_ids': [active_id], 'default_model': 'res.partner', 'default_res_id': active_id}"
+                        icon="fa-envelope">
+                        <field string="Mails to" name="mails_to" widget="statinfo"/>
+                    </button>
+                    <button class="oe_inline oe_stat_button" type="action"
+                        name="%(action_mails)d"
+                        context="{'search_default_author_id': active_id, 'default_model': 'res.partner', 'default_res_id': active_id}"
+                        icon="fa-envelope-o">
+                        <field  string="Mails from" name="mails_from" widget="statinfo"/>
+                    </button>
+                </xpath>
+            </field>
+        </record>
+    </data>
+</openerp>

res_partner_mails_count/tests/__init__.py

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+# -*- coding: utf-8 -*-
+
+from . import test_mail

res_partner_mails_count/tests/test_mail.py

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+# -*- coding: utf-8 -*-
+
+from openerp.tests.common import TransactionCase
+
+
+class test_message_count(TransactionCase):
+    post_install = True
+    def test_count(self):
+        new_partner1 = self.env['res.partner'].sudo().create({'name': 'rpmc Test Partner one', 'email': 'tt@tt', 'notify_email': 'always'})
+        new_partner2 = self.env['res.partner'].sudo().create({'name': 'rpmc Test Partner two', 'email': 'rr@rr', 'notify_email': 'always'})
+        self.assertEqual(new_partner1.mails_to, 0, 'rpmc: new partner have mails_to != 0')
+        mail_compose = self.env['mail.compose.message']
+        compose = mail_compose.with_context(
+            {
+                'default_composition_mode': 'comment',
+            }).create(
+            {
+                'subject': 'test subj',
+                'body': 'test body',
+                'partner_ids': [(4, new_partner2.id)],
+                'email_from': 'tt@tt',
+                'author_id': new_partner1.id
+            })
+        compose.send_mail()
+        self.assertEqual(new_partner1.mails_to, 0)
+        self.assertEqual(new_partner1.mails_from, 1, 'rpmc: one message but mails_from != 1')
+        self.assertEqual(new_partner2.mails_to, 1, 'rpmc: one message but mails_to != 1')
+        self.assertEqual(new_partner2.mails_from, 0)
+        compose.send_mail()
+        self.assertEqual(new_partner1.mails_to, 0)
+        self.assertEqual(new_partner1.mails_from, 2, 'rpmc: one message but mails_from != 2')
+        self.assertEqual(new_partner2.mails_to, 2, 'rpmc: one message but mails_to != 2')
+        self.assertEqual(new_partner2.mails_from, 0)

res_partner_mails_count/views/res_partner_mails_count.xml

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+<?xml version="1.0" encoding="UTF-8"?>
+<openerp>
+    <data>
+        <record id="action_mails" model="ir.actions.client">
+            <field name="name">Mails</field>
+            <field name="tag">mail.wall</field>
+            <field name="res_model">mail.message</field>
+            <field name="context">{
+                'ignore_search_model': True,
+            }</field>
+            <field name="help" type="html">
+                <p>
+                    Mails not found. Probably, they exist, but you don't have access.
+                </p>
+            </field>
+        </record>
+    </data>
+</openerp>